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Full text of "Newton's Principia : the mathematical principles of natural philosophy"

MATH.-STAT. 




SIM ISAAC MIBWf OM 



NEWTON S PRINCIPIA. 



THE 



MATHEMATICAL PRINCIPLES 



OF 



NATURAL PHILOSOPHY, 

BY SIR ISAAC NEWTON; 

TRANSLATED INTO ENGLISH BY ANDREW MOTTE. 



TO WHICH IS ADDKTV 



NEWTON S SYSTEM OF THE WORLD ; 

With a Portrait taken from the Bust in the Royal Observatory at Greenwich. 

FIRST AMERICAN EDITION, CAREFULLY REVISED AND CORRECTED, 

WITH A LIFE OF THE AUTHOR, BY PI. W. CHITTENDEN, M. A., &e. 



NEW-YORK 

PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET. 




p*- 



Kntered according to Act of Congress, in the year 1846, by 

DANIEL ADEE. 
3!Ltht Clerk s Office ut tiie Southern Oisli:ct Court of New-York. 



TWuey * Lockwoof, Stom 
16 Spruce St. N. Y. 



DEDICATION. 



TO THE 

TEACHERS OF THE NORMAL SCHOOL 

OF THE STATE OF NEW-YORK. 

GENTLEMEN ! 

A stirring freshness in the air, and ruddy streaks upon the 
horizon of the moral world betoken the grateful dawning of a new 
ora. The days of a drivelling instruction are departing. With 
us is the opening promise of a better time, wherein genuine man 
hood doing its noblest work shall have adequate reward. 
TEACHER is the highest and most responsible office man can fill. 
Its dignity is, and will yet be held commensurate with its duty 
a duty boundless as man s intellectual capacity, and great as his 
moral need a duty from the performance of which shall emanate 
an influence not limited to the now and the here, but which surely 
will, as time flows into eternity and space into infinity, roll up, a 
measureless curse or a measureless blessing, in inconceivable 
swellings along the infinite curve. It is an office that should be 
esteemed of even sacred import in this country. Ere long a hun 
dred millions, extending from the Atlantic to the Pacific, from 
Baffin s Bay to that of Panama, shall call themselves American 
citizens. What a field for those two master-passions of the hu 
man soul the love of Rule, and the love of Gain ! How shall 
our liberties continue to be preserved from the graspings of Am 
bition and the corruptions of Gold ? Not by Bills of Rights 



4 DEDICATION. 

Constitutions, and Statute Books ; but alone by the rightly culti 
vated hearts and heads of the PEOPLE. They must themselves 
guard the Ark. It is yours to tit them for the consecrated 
charge. Look well to it : for you appear clothed in the majesty 
of great power ! It is yours to fashion, and to inform , to save, 
and to perpetuate. You are the Educators of the People : you 
are the prime Conservators of the public weal. Betray your 
trust, and the sacred fires would go out, and the altars crumble 
into dust : knowledge become lost in tradition, and Christian no 
bleness a fable ! As you, therefore, are multiplied in number, 
elevated in consideration, increased in means, and fulfill, well and 
faithfully, all the requirements of true Teachers, so shall our fa 
voured land lift up her head among the nations of the earth, and 
call herself blessed. 

In conclusion, Gentlemen, to you, as the conspicuous leaders 
in the vast and honourable labour of Educational Helbrm, ana 
Popular Teaching, the First American Edition of the PRINCIPIA ol 
Newton the greatest w r ork of the greatest Teacher is most 
respectfully dedicated. 

N. W. CHITTENDEN. 



INTRODUCTION TO THE AMERICAN EDITION. 



THAT the PRINCIPIA of Newton should have remained so gen 
erally unknown in this country to the present day is a somewhat 
remarkable fact ; because the name of the author, learned with 
the very elements of science, is revered at every hearth-stone 
where knowledge and virtue are of chief esteem, while, abroad, 
in all the high places of the land, the character which that name 
recalls is held up as the noblest illustration of what MAN may be, 
and may do, in the possession and manifestation of pre-eminent 
intellectual and moral worth ; because the work is celebrated, not 
only in the history of one career and one mind, but in the history 
of all achievement and human reason itself; because of the spirit 
of inquiry, which has been aroused, and which, in pursuing its 
searchings, is not always satisfied with stopping short of the foun 
tain-head of any given truth ; and, finally, because of the earnest 
endeavour that has been and is constantly going on, in many 
sections of the Republic, to elevate the popular standard of edu 
cation and give to scientific and other efforts a higher and a 
better aim. 

True, the PRINCIPIA has been hitherto inaccessible to popular 
use. A few copies in Latin, and occasionally one in English may 
be found in some of our larger libraries, or in the possession of 
some ardent disciple of the great Master. But a d^ad language 
in the one case, and an enormous price in both, particularly in 
that of the English edition, have thus far opposed very sufficient 
obstacles to the wide circulation of the work. It is now, how 
ever, placed within the reach of all. And in performing this la 
bour, the utmost care has been taken, by collation, revision, and 
otherwise, to render the First American Edition the most accurate 
and beautiful in our language. u Le plus beau monument que 
l ? on puisse clever a la gloire de Newton, c est une bonne edition 
de ses ouvrages :" and a monument like unto that we would here 



V: INTRODUCTION TO 

set up. The PRINCIPIA, above all, glows with the immortality of 
a transcendant mind. Marble and brass dissolve and pass away ; 
but the true creations of genius endure, in time and beyond time, 
forever : high upon the adamant of the indestructible, they send 
forth afar and near, over the troublous waters of life, a pure, un 
wavering, quenchless light whereby the myriad myriads of barques, 
richly laden with reason, intelligence and various faculty, are 
guided through the night and the storm, by the beetling shore 
and the hidden rock, the breaker and the shoal, safely into havens 
calm and secure. 

To the teacher and the taught, the scholar and the student, the 
devotee of Science and the worshipper of Truth, the PRINCIPIA 
must ever continue to be of inestimable value. If to educate 
means, not so much to store the memory with symbols and facts, 
as to bring forth the faculties of the soul and develope them to the 
full by healthy nurture and a hardy discipline, then, what so effec 
tive to the accomplishment of that end as the study of Geometri 
cal Synthesis ? The Calculus, in some shape or other, is, indeed, 
necessary to the successful prosecution of researches in the higher 
branches of philosophy. But has not the Analytical encroached 
upon the Synthetical, and Algorithmic Formulae been employed 
when not requisite, either for the evolution of truth, or even its 
apter illustration ? To each method belongs, undoubtedly, an 
appropriate use. Newton, himself the inventor of Fluxions, 
censured the handling of Geometrical subjects by Algebraical 
calculations ; and the maturest opinions which he expressed were 
additionally in favour of the Geometrical Method. His prefer 
ence, so strongly marked, is not to be reckoned a mere matter oi 
taste ; and his authority should bear with preponderating weight 
upon the decision of every instructor in adopting what may be 
deemed the best plan to insure the completes! mental develop 
ment. Geometry, the vigorous product of remote time ; blended 
with the earliest aspirations of Science and the earliest applica 
tions of Art ; as well in the measures of music as in the move 
ment of spheres ; as wholly in the structure of the atom as in that 
of the world; directing MOTION and shaping APPEARANCE; in a 
wonl, *t the moulding of the created all, is, in comprehensive 



THE AMERICAN EDITION. Vll 

view, the outward form of that Inner Harmony of which and in 
which all things are. Plainly, therefore, this noble study has 
other and infinitely higher uses than to increase the power of ab 
straction. A more general and thorough cultivation of it should 
oe strenuously insisted on. Passing from the pages of Euclid or 
Legendre, might not the student be led, at the suitable time, to 
those of the PRINCIPIA wherein Geometry may be found in varied 
use from the familiar to the sublime ? The profoundest and the 
happiest results, it is believed, would attend upon this enlargement 
of our Educational System. 

Let the PRINCIPIA, then, be gladly welcomed into every Hall 
where a TRUE TEACHER presides. And they who are guided to 
the diligent study of this incomparable work, who become 
strengthened by its reason, assured by its evidence, and enlight 
ened by its truths, and who rise into loving communion with the 
great and pure spirit of its author, will go forth from the scenes 
of their pupilage, and take their places in the world as strong- 
minded, right-hearted men such men as the Theory of our 
Government contemplates and its practical operation absolutely 
demands. 



LIFE OF 

SIE ISAAC NEWTON. 



Nec fas est proprius mortal? attingere Divos. HALLEY. 



FROM the thick darkness of the middle ages man s struggling 
spirit emerged as in new birth ; breaking out of the iron control 
of that period ; growing strong and confident in the tug and din 
of succeeding conflict and revolution, it bounded forwards and 
upwards with resistless vigour to the investigation of physical and 
moral truth ; ascending height after height ; sweeping afar over 
the earth, penetrating afar up into the heavens ; increasing in en 
deavour, enlarging in endowment ; every where boldly, earnestly 
out-stretching, till, in the AUTHOR of the PRINCIPIA, one arose, 
who, grasping the master-key of the universe and treading its 
celestial paths, opened up to the human intellect the stupendous 
realities of the material world, and, in the unrolling of its harmo 
nies, gave to the human heart a new song to the goodness, wis 
dom, and majesty of the all-creating, all-sustaining, all-perfect 
God. 

Sir Isaac Newton, in whom the rising intellect seemed to attain, 
as it were, to its culminating point, was born on the 25th of De 
cember, O. S. 1642 Christmas day at Woolsthorpe, in the 
parish of Colsterworth, in Lincolnshire. His father, John New 
ton, died at the age of thirty-six, and only a few months after his 
marriage to Harriet Ayscough, daughter of James Ayscough, oi 
Rutlandshire. Mrs. Newton, probably wrought upon by the 
early loss of her husband, gave premature birth to her only and 
posthumous child, of which, too, from its extreme diminutiveness, 
she appeared likely to be soon bereft. Happily, it was otherwise 
decreed ! The tiny infant, on whose little lips the breath of life 



10 LIFE OF SIR ISAAC NEWTON. 

so doubtingly hovered, lived ; lived to a vigorous maturity, to a 
hale old age ; lived to become the boast of his country, the won 
der of his time, and the "ornament of his srjecies." 

Beyond the grandfather, Robert Newton, the descent of Sir 
Isaac cannot with certainty be traced. Two traditions were held 
in the family : one, that they were of Scotch extraction ; the 
other, that they came originally from Newton, in Lancashire, 
dwelling, for a time, however, at Westby, county of Lincoln, be 
fore the removal to and purchase of Woolsthorpe about a hundred 
years before this memorable birth. 

The widow Newton was left with the simple means of a com 
fortable subsistence. The Woolsthorpe estate together with 
small one which she possessed at Sewstern, in Leicestershire, yield 
ed her an income of some eighty pounds ; and upon this limited sum, 
she had to rely chiefly for the support of herself, and the educa 
tion of her child. She continued his nurture for three years, 
when, marrying again, she confided the tender charge to the care 
of her own mother. 

Great genius is seldom marked by precocious development ; 
and young Isaac, sent, at the usual age, to two day schools at 
Skillington and Stoke, exhibited no unusual traits of character. 
In his twelfth year, he was placed at the public school at Gran- 
tham, and boarded at the house of Mr. Clark, an apothecary. 
But even in this excellent seminary, his mental acquisitions con 
tinued for a while unpromising enough : study apparentlv had no 
charms for him ; he was very inattentive, and ranked low in the 
school. One day, however, the boy immediately above our seem 
ingly dull student gave him a severe kick in the stomach ; Isaac, 
deeply affected, but with no outburst of passion, betook himself, 
with quiet, incessant toil, to his books ; he quickly passed above 
the offending classmate ; yet there he stopped not ; the strong 
spirit was, for once and forever, awakened, and, yielding to itb 
noble impulse, he speedily took up his position at the head of all. 

His peculiar character began now rapidly to unfold itself. 
Close application grew to be habitual. Observation alternated 
with reflection. " A sober, silent, thinking lad," yet, the wisest 
and the kindliest, the indisputable leader of his fellows. Gener- 



LIFE OF SIR ISA VC NEWTON. 11 

osity, modesty, and a love of truth distinguished him then as ever 
afterwards. He did not often join his classmates in play ; but he 
would contrive for them various amusements of a scientific kind. 
Paper kites he introduced ; carefully determining their best form 
and proportions, and the position and number of points whereby 
to attach the string. He also invented paper lanterns ; these 
served ordinarily to guide the way to school in winter mornings, 
but occasionally for quite another purpose ; they were attached to 
the tails of kites in a dark night, to the dismay of the country people 
dreading portentous comets, and to the immeasureable delight ol 
his companions. To him, however, young as he was, life seemed 
to have become an earnest thing. When not occupied with his 
studies, his mind would be engrossed with mechanical contrivances ; 
now imitating, now inventing. He became singularly skilful in the 
use of his little saws, hatchets, hammers, and other tools. A 
windmill was erected near Grantham ; during the operations ol 
the workmen, he was frequently present ; in a short time, he had 
completed a perfect working model of it, which elicited general 
admiration. Not content, however, with this exact imitation, he 
conceived the idea of employing, in the place of sails, animal power , 
and, adapting the construction of his mill accordingly, he enclosed 
in it a mouse, called the miller, and which by acting on a sort ot 
treadvvheel, gave motion to the machine. He invented, too, a 
mechanical carriage having four wheels, and put in motion with 
a handle worked by the person sitting inside. The measurement 
of time early drew his attention. He h rst constructed a water 
clock, in proportions somewhat like an old-fashioned house clock. 
The index of the dial plate was turned by a piece of wood acted 
upon by dropping water. This instrument, though long used by 
himself, and by Mr. Clark s family, did not satisfy his inquiring 
mind. His thoughts rose to the sun ; and, by careful and oft-re 
peated observations of the solar movements, he subsequently 
formed many dials. One of these, named Isaac s dial, was the 
accurate result of years labour, and was frequently referred to 
for the hour of the day by the country people. 

May we not discern in these continual efforts the diligent re 
search^ the patient meditation, the aspiring glance, and the energy 



12 LIFE OF SIR ISAAC NEWTON. 

of discovery the stirring elements of that wondrous spirit, 
which, clear, calm, and great, moved, in after years, through 
deep onward through deep of Nature s mysteries, unlocking her 
strongholds, dispelling darkness, educing order everywhere si 
lently conquering. 

Newton had an early and decided taste for drawing. Pictures, 
taken sometimes from copies, but often from life, and drawn, 
coloured and framed by himself, ornamented his apartment. He 
was skilled also, in poetical composition, " excelled in making 
verses ;" some of these were borne in remembrance and repeated, 
seventy years afterward, by Mrs. Vincent, for whom, in early 
youth, as Miss Storey, he formed an ardent attachment. She 
was the sister of a physician resident near Woolsthorpe ; but 
Newton s intimate acquaintance with her began at Grantham. 
where they were both numbered among the inmates of the same 
house. Two or three years younger than himself, of great per 
sonal beauty, and unusual talent, her society afforded him the 
greatest pleasure ; and their youthful friendship, it is believed, 
gradually rose to a higher passion ; but inadequacy of fortune 
prevented their union. Miss Storey was afterwards twice mar 
ried ; Newton, never; his esteem for her continued unabated 
during life, accompanied by numerous acts of attention and 
kindness. 

In 1656, Newton s mother was again left a widow r , and took 
up her abode once more at Woolsthorpe. He was now fifteen 
years of age, and had made great progress in his studies ; but she, 
desirous of his help, and from motives of economy, recalled him 
from school. Business occupations, however, and the manage 
ment of the farm, proved utterly distasteful to him. When sent to 
Grantham Market on Saturdays, he would betake himself to his 
former lodgings in the apothecary s garret, where some of Mr. 
Clark s old books employed his thoughts till the aged and trust 
worthy servant had executed the family commissions and announced 
the necessity of return : or, at other times, our young philosopher 
would seat himself under a hedge, by the wayside, and continue 
his studies till the same faithful personage proceeding alone to 
the town and completing the day s business stopped as he re- 



LIFE OF SIR ISAAC NEWTON, 13 

turned. The more immediate affairs of the farm received no 
better attention. In fact, his passion for study grew daily more 
absorbing, and his dislike for every other occupation more in 
tense. His mother, therefore, wisely resolved to give him all the 
advantages which an education could confer. He was sent back 
to Grantham school, where he remained for some months in busy 
preparation for his academical studies. At the recommendation 
of one of his uncles, who had himself studied at Trinity College, 
Cambridge, Newton proceeded thither, and was duly admitted. 
on the 5th day of June 1660, in the eighteenth year of his age. 

The eager student had now entered upon a new and wider 
field ; and we find him devoting himself to the pursuit of know 
ledge with amazing ardour and perseverance. Among other sub 
jects, his attention was soon drawn to that of Judicial Astrology 
He exposed the folly of this pseudo-science by erecting a figure 
with the aid of one or two of the problems of Euclid ; and thus 
began his study of the Mathematics. His researches into this 
science were prosecuted with unparallelled vigour and success. 
Regarding the propositions contained in Euclid as self-evident 
truths, he passed rapidly over this ancient system a step which 
he afterward much regretted and mastered, without further pre 
paratory study, the Analytical Geometry of Descartes. Wallis s 
Arithmetic of Infinites, Saunderson s Logic, and the Optics of 
Kepler, he also studied with great care ; writing upon them 
many comments ; and, in these notes on Wallis s work was un 
doubtedly the germ of his fluxionary calculus. His progress was 
so great that he found himself more profoundly versed than his tutor 
in many branches of learning. Yet his acquisitions were not 
gotten with the rapidity of intuition ; but they were thoroughly 
made and firmly secured. Quickness of apprehension, or Intel 
lectual nimbleness did not belong to him. He saw too far : his, 
insight was too deep. He dwelt fully, cautiously upon the least 
subject ; while to the consideration of the greatest, he brought a 
massive strength joined with a matchless clearness, that, regard 
less of the merely trivial or unimportant, bore with unerring sa 
gacity upon the prominences of the subject, and, grappling with 
its difficulties, rarely failed to surmount them. 



14 LIFE OF SIR ISAAC NEWTON 

His early and fast friend, Dr. Barrow in compass of inven 
tion only inferior to Newton who had been elected Professor 
of Greek in the University, in 1660, was made Lucasian Profes 
sor of Mathematics in 1663, and soon afterward delivered his 
Optical Lectures : the manuscripts of these were revised by New 
ton, and several oversights corrected, and many important sug 
gestions made by him ; but they were not published till 1669. 

In the year 1665, he received the degree of Bachelor of Arts ; 
and, in 1666, he entered upon those brilliant and imposing dis 
coveries which have conferred inappreciable benefits upon science, 
and immortality upon his own name. 

Newton, himself, states that he was in possession of his Method 
of Fluxions, " in the year 1666, or before." Infinite quantities 
had long been a subject of profound investigation ; among the 
ancients by Archimedes, and Pappus of Alexandria ; among the 
moderns by Kepler, Cavaleri, Roberval, Fermat and Wallis. 
With consummate ability Dr. Wallis had improved upon the la- 
hours of his predecessors : with a higher power, Newton moved 
forwards from where Wallis stopped. Our author first invented 
his celebrated BINOMIAL THEOREM. And then, applying this 
Theorem to the rectification of curves, and to the determination 
of the surfaces and contents of solids, and the position of their 
centres of gravity, he discovered the general principle of deducing 
the areas of curves from the ordinate, by considering the area as 
a nascent quantity, increasing by continual fluxion in the propor 
tion of the length of the ordinate, and supposing the abscissa 
to increase uniformly in proportion to the time. Regarding lines 
as generated by the motion of points, surfaces by the motion of 
lines, and solids by the motion of surfaces, and considering that 
the ordinates, abscissae, &c., of curves thus formed, vary accord 
ing to a regular law depending on the equation of the curve, 
he deduced from this equation the velocities with which these 
quantities are generated, and obtained by the rules of infinite 
series, the ultimate value required. To the velocities with which 
every line or quantity is generated, he gave the name of FLUX 
IONS, and to the lines or quantities themselves, that of FLUENTS. 
A discovery that successively baffled the acutest and strongest 



LIFE OF SIR ISAAC NEWTON. 15 

intellects : that, variously modified, has proved of incalculable 
service in aiding to develope the most abstruse and the highest 
ruths in Mathematics and Astronomy : and that was of itself 
enough to render any name illustrious in the crowded Annals of 
Science. 

At this period, the most distinguished philosophers were direct 
ing all their energies to the subject of light and the improvement 
of the refracting telescope. Newton, having applied himself to 
the grinding of "optic glasses of other figures than spherical," ex 
perienced the impracticability of executing such lenses ; and con 
jectured that their defects, and consequently those of refracting 
telescopes, might arise from some other cause than the imperfect 
convergency of rays to a single point. He accordingly "procured 
a triangular glass prism to try therewith the celebrated phenom 
ena of colours." His experiments, entered upon with zeal, and 
conducted with that industry, accuracy, and patient thought, for 
which he was so remarkable, resulted in the grand conclusion, 
that LIGHT WAS NOT HOMOGENEOUS, BUT CONSISTED OF RAYS, 

SOME OF WHICH WERE MORE REFRANGIBLE THAN OTHERS. This 

profound and beautiful discovery opened up a new era in the 
History of Optics. As bearing, however, directly upon the construc 
tion of telescopes, he saw that a lens refracting exactly like a prism 
would necessarily bring the different rays to different foci, at 
different distances from the glass, confusing and rendering the 
vision indistinct. Taking for granted that all bodies produced 
spectra of ^ jtial length, he dismissed all further consideration of 
the refracting instrument, and took up the principle of reflection. 
Rays of all colours, he found, were reflected regularly, so that the 
angle of reflection was equal to the angle of incidence, and hence 
he concluded that ojitical instruments might be brought to any 
degree of perfection imaginable, provided reflecting specula of 
the requisite figure and finish could be obtained. At this stage 
of his optical researches, he was forced to leave Cambridge on 
account of the plague which was then desolating England. 

He retired to Woolsthorpe. The old manor-house, in which he 
was born, was situated in a beautiful little valley, on the west side 
of the river Witham ; and here in the quiet home of his boyhood, 

2 



16 LIFE OF SIR ISAAC NEWTON. 

he passed his days in serene contemplation, while the stalking 
pestilence was hurrying its tens of thousands into undistinguisha 
ble graves. 

Towards the close of a pleasant day in the early autumn of 
1666, he was seated alone beneath a tree, in his garden, absorbed 
in meditation. He was a slight young man ; in the twenty-fourth 
year of his age ; his countenance mild and full of thought. For 
a century previous, the science of Astronomy had advanced with 
rapid strides. The human mind had risen from the gloom and 
bondage of the middle ages, in unparalleled vigour, to unfold the 
system, to investigate the phenomena, and to establish the laws 
of the heavenly bodies. Copernicus, Tycho Brahe, Kepler, 
Galileo, and others had prepared and lighted the way for him 
who was ta give to their labour its just value, and to their genius 
its true lustre. At his bidding isolated facts were to take order 
as parts of one harmonious whole, and sagacious conjectures grow 
luminous in the certain splendour of demonstrated truth. And 
this ablest man had come was here. His mind, familiar with 
the knowledge of past effort, and its unequalled faculties develop 
ed in transcendant strength, was now moving on to the very 
threshold of Its grandest achievement. Step by step the untrod 
den path was measured, till, at length, the entrance seemed dis 
closed, and the tireless explorer to stand amid the first opening 
wonders of the universe. 

The nature of gravity that mysterious power which causes 
all bodies to descend towards the centre of the earth had, in 
deed, dawned upon him. And reason busily united link to link 
of that chain which was yet to be traced joining the least to the 
vastest, the most remote to the nearest, in one harmonious bond. 
From the bottoms of the deepest caverns to the summits of the 
highest mountains, this power suffers no sensible change : may not 
its action, then, extend to the moon ? Undoubtedly : and furthei 
reflection convinced him that such a power might be .sufficient for 
retaining that luminary in her orbit round the earth. But, though 
this power suffers no sensible variation, in the little change of 
distance from the earth s centre, at which we may place our- 
. lves, yet, at the distance of the moon, :miy not its force undergo 



LIFE OF SIR ISAAC NEWTON. 17 

more or less diminution ? The conjecture appeared most proba 
ble : and, in order to estimate what the degree of diminution 
might be, he considered that if the moon be retained in her orbit 
by the force of gravity, the primary planets must also be carried 
round the sun by the like power; and, by comparing the periods 
of the several planets with their distances from the sun, he found 
that, if they were held in their courses by any power like gravity, 
its strength must decrease in the duplicate proportion of the in 
crease of distance. In forming this conclusion, he supposed the 
planets to move in perfect circles, concentric to the sun. Now 
was this the law of the moon s motion ? Was such a force, em 
anating from the earth and directed to the moon, sufficient, when 
diminished as the square of the distance, to retain her in her 
orbit ? To ascertain this master-fact, he compared the space 
through which heavy bodies fall, in a second of time, at a given 
distance from the centre of the earth, namely, at its surface, with 
the space through which the moon falls, as it were, to the earth, 
in the same time, while revolving in a circular orbit. He was 
absent from books ; and, therefore, adopted, in computing the 
earth s diameter, the common estimate of sixty miles to a degree 
of latitude as then in use among geographers and navigators. 
The result of his calculations did not, ot course, answer his ex 
pectations ; hence, he concluded that some other cause, beyond the 
reach of observation analogous, perhaps, to the vortices of Des 
cartes joined its action to that of the power of gravity upon the 
rnooil. Though by no means satisfied, he yet abandoned awhile 
further inquiry, and remained totally silent upon the subject. 

These rapid marches in the career of discovery, combined with 
the youth of Newton, seem to evince a penetration the most 
lively, and an invention the most exuberant. But in him there 
was a conjunction of influences as extraordinary as fortunate. 
Study, unbroken, persevering and profound carried on its inform 
ing and disciplining work upon a genius, natively the greatest, 
and rendered freest in its movements, and clearest in its vision, 
through the untrammelling and enlig} tenirig power of religion. 
And, in this happy concurrence, are to be sought the elements of 
those amazing abilities, which, grasping, with equal facility, the 



18 LIFE OF SIR ISAAC NEWTON. 

minute and the stupendous, brought these successively to light, 
and caused science to make them her own. 

In 1667, Newton was made a Junior Fellow ; and, in the year 
following, he took his degree of Master of Arts, and was appoint 
ed to a Senior Fellowship. 

On his return to Cambridge, in 1668, he resumed his optical 
labours. Having thought of a delicate method of polishing metal, 
he proceeded to the construction of his newly projected reflect 
ing telescope ; a small specimen of which he actually made with 
his own hands, It was six inches long ; and magnified about 
forty times ; a power greater than a refracting instrument of six 
feet tube could exert with distinctness. Jupiter, with his four 
satellites, and the horns, or moon-like phases of Venus were 
plainly visible through it. THIS WAS THE FIRST REFLECTING 

TELESCOPE EVER EXECUTED AND DIRECTED TO THE HEAVENS. 

He gave an account of it, in a letter to a friend, dated February 23d, 
1668-9 a letter which is also remarkable for containing the firs 
allusion to his discoveries " concerning the nature of light." En 
couraged by the success of his first experiment, he again executed 
with his own hands, not long afterward, a second and superior 
instrument of the same kind. The existence of this having come 
to the knowledge of the Royal Society of London, in 1671, they 
requested it of Newton for examination. He accordingly sent it 
to them, It excited great admiration; it was shown to the king* 
a drawing and description of it was sent to Paris ; and the tele- 
scope itself was carefully preserved in the Library of the Society. 
Newton lived to see his invention in public use, and of eminent 
service in the cause of science. 

In the spring of 1669, he wrote to his friend Francis Aston, 
Esq., then about setting out on his travels, a letter of advice and 
directions, it was dated May 18th, and is interesting as exhibit 
ing some of the prominent features in Newton s character. 
Thus : 

" Since in your letter you give me so much liberty of spending 
my judgment about what may be to your advantage in travelling, 
1 shall do it more freely than perhaps otherwise would have been 
decent, Fir, c t, then, I will lay down some general rules, most of 



LIFE OF SIR ISAAC NEWTON. 19 

which, I bolieA e, you have considered already ; but if any of 
them be new to you, they may excuse the rest ; if none at all, 
yet is my punishment more in writing than yours in reading. 

"When you come into any fresh company. 1. Observe their 
humours. 2. Suit your own carriage thereto, by which insinua 
tion you will make their converse more free and open. 3. Let 
your discourse be more in queries and doubtings than peremptory 
assertions or disputings, it being the design of travellers to learn, 
not to teach. Besides, it will persuade your acquaintance that 
you have the greater esteem of them, and so make them more 
ready to communicate what they know to you ; whereas nothing 
sooner occasions disrespect and quarrels than peremptoriness. 
You will find little or no advantage in seeming wiser or much 
more ignorant than your company. 4. Seldom discommend any 
thing though never so bad, or do it but moderately, lest you be 
unexpectedly forced to an unhandsome retraction. It is safer to 
commend any thing more than it deserves, than to discommend 
a thing so much as it deserves; for commendations meet not 
so often with oppositions, or, at least, are not usually so ill re 
sented by men that think otherwise, as discommendations ; and 
you will insinuate into men s favour by nothing sooner than seem 
ing to approve and commend what they like ; but beware o 
doing it by comparison. 5. If you be affronted, it is better, in c 
foreign country, to pass it by in silence, and with a jest, though 
with some dishonour, than to endeavour revenge ; for, in the first 
case, your credit s ne er the worse when you return into England, 
or come into other company that have not heard of the quarrel. 
But, in the second case, you may bear the marks of the quarrel 
while you live, if you outlive it at all. But, if you find yoursell 
unavoidably engaged, tis best, I think, if you can command your 
passion and language, to keep them pretty evenly at some certain 
moderate pitch, not much heightening them to exasperate your 
adversary, or provoke his friends, nor letting them grow overmuch 
dejected to make him insult. In a word, if you can keep reason 
above passion, that and watchfulness will be your best defendants. 
To which purpose you may consider, that, though such excuses 
is this He provok t me so much I could not forbear may pass 



20 LIFE OF SIR ISAAC NEWTON. 

among friends, yet amongst strangers they are insignificant, ina 
only argue a traveller s weakness. 

" To these I may add some general heads for inquiries or ob 
servations, such as at present I can think on. As, 1. To observe 
the policies, wealth, and state affairs of nations, so far as a soli- 
f ary traveller may conveniently do. 2. Their impositions upon 
all sorts of people, trades, or commodities, that are remarkable. 
3. Their laws and customs, how far they differ from ours. 4. 
Their trades and arts wherein they excel or come short of us in 
England. 5. Such fortifications as you shall meet with, their 
fashion, strength, and advantages for defence, and other such mili 
tary affairs as are considerable. 6. The power and respect be 
longing to their degrees of nobility or magistracy. 7. It will not 
be time misspent to make a catalogue of the names and excellen 
cies of those men that are most wise, learned, or esteemed in any 
nation. 8. Observe the mechanism and manner of guiding ships. 

9. Observe the products of Nature in several places, especially in 
mines, with the circumstances of mining and of extracting metals 
or minerals out of their ore, and of refining them ; and if you 
meet with any transmutations out of their own species into 
another (as out of iron into copper, out of any metal into quick 
silver, out of one salt into another, or into an insipid body, &c.), 
those, above all, will be worth your noting, being the most lucif- 
erous, and many times lucriferous experiments, too, in philosophy. 

10. The prices of diet and other things. 11. And the staple 
commodities of places. 

" These generals (such as at present I could think of), if they 
will serve for nothing else, yet they may assist you in drawing up 
a model to regulate your travels by. As for particulars, these that 
follow are all that 1 can now think of, viz. ; whether at Schem- 
nitium, in Hungary (where there are mines of gold, copper, iron, 
vitriol, antimony, &c.). they change iron into copper by dissolving 
t in a vitriolate water, which they find in cavities of rocks in the 
mines, and then melting the slimy solution in a stroi ig fire, which 
in the cooling proves copper. The like is said to be done in other 
places, which I cannot now remember ; perhaps, too, it may be 
lone in Italy. For about twenty or thirty years agone there was 



LIFE OF SIR ISAAC NEWTON, 21 

a certain vitriol came from thence (called Roman vitriol), but of 
a nobler virtue than that which is now called by that name ; 
which vitriol is not now to be gotten, because, perhaps, they make 
a greater gain by some such trick as turning iron into copper 
with it than by selling it. 2. Whether, in Hungary, Sclavonia, 
Bohemia, near the town Eila, or at the mountains of Bohemia 
near Silesia, there be rivers whose waters are impregnated with 
gold ; perhaps, the gold being dissolved by some corrosive water 
like aqua regis, and the solution carried along with the stream, 
that runs through the mines. And whether the practice of laying 
mercury in the rivers, till it be tinged with gold, and then strain 
ing the mercury through leather, that the gold may stay behind, 
be a secret yet, or openly practised. 3. There is newly con 
trived, in Holland, a mill to grind glasses plane withal, and I 
think polishing them too ; perhaps it will be worth the while to see 

it. 4. There is in Holland one Borry, who some years since 

was imprisoned by the Pope, to have extorted from him secrets 
(as I am told) of great worth, both as to medicine and profit, but 
he escaped into Holland, where they have granted him a guard. 
I think he usually goes clothed in green. Pray inquire what you 
can of him, and whether his ingenuity be any profit to the Dutch. 
You may inform yourself whether the Dutch have any tricks to 
keep their ships from being all worm-eaten in their voyages to 
the Indies. Whether pendulum clocks do any service in finding 
out the longitude, &c. 

" I am very weary, and shall not stay to part with a long 
compliment, only I wish you a good journey, and God be with 
you." 

It was not till the month of June, 1669, that our author made 
known his Method of Fluxions. He then communicated the 
work which he had composed upon the subject, and entitled, 
ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS, 
to his friend Dr. Barrow. The latter, in a letter dated 20th of the 
same month, mentioned it to Mr. Collins, and transmitted it to 
him, on the 31st of July thereafter. Mr. Collins greatly approv> 
ed of the work ; took a copy of it ; and sent the original back 
to Dr. Barrow. During the same and the two following years, Mr 



< LIFE OF SIR ISAAC NEWTON. 

Collins, by his extensive correspondence, spread the knowledge 
of this discovery among the mathematicians in England, Scotland, 
France, Holland and Italy. 

Dr. Barrow, having resolved to devote himself to Theology, 
resigned the Lucasian Professorship of Mathematics, in 1669, in 
favour of Newton, who accordingly received the appointment to 
the vacant chair. 

During the years 1669, 1670, and 1671, our author, as such 
Professor, delivered a course of Optical Lectures. Though these 
contained his principal discoveries relative to the different re- 
frangibility of light, yet the discoveries themselves did not be 
come publicly known, it seems, till he communicated them to the 
Royal Society, a few weeks after being elected a member there 
of, in the spring of 1671-2. He now rose rapidly in reputation, 
and was soon regarded as foremost among the philosophers of the 
age. His paper on light excited the deepest interest in the Royal 
Society, who manifested an anxious solicitude to secure the author 
from the " arrogations of others," and proposed to publish his 
discourse in the monthly numbers in which the Transactions were 
given to the world. Newton, gratefully sensible of these expres 
sions of esteem, willingly accepted of the proposal for publication. 
He gave them also, at this time, the results of some further ex 
periments in the decomposition and re-composition of light : that 
the same degree of refrangibility always belonged to the same 
colour, and the same colour to the same degree of refrangibility : 
that the seven different colours of the spectrum were original, or 
simple, and that whiteness^ or white light was a compound of all 
these seven colours. 

The publication of his new doctrines on light soon called forth 
violent opposition as to their soundness. Hooke and Huygens 
men eminent for ability and learning were the most conspicuous 
of the assailants. And though Newton effectually silenced all his 
adversaries, yet he felt the triumph of little gain in comparison 
.vith the loss his tranquillity had sustained. He subsequently re- 
narked in allusion to this controversy and to one with whom 
he was destined to have a longer and a bitterer conflict " I was 
so persecuted with discussions arising from the publication of m v 



LIFE OF SIR ISAAC NEWTON. 23 

theory ot light, that I blamed my own imprudence for parting 
with so substantial a blessing as rny quiet to run after a shadow. 7 

In a communication to Mr. Oldenburg, Secretary of the Royal 
Society, in 1672, our author stated many valuable suggestions re 
lative to the construction of REFLECTING MICROSCOPES which he 
considered even more capable of improvement than telescopes. 
He also contemplated, about the same time, an edition of Kirick- 
huysen s Algebra, with notes and additions; partially arranging, 
as an introduction to the work, a treatise, entitled, A Method of 
Fluxions ; but he finally abandoned the design. This treatise, 
however, he resolved, or rather consented, at a late period of his 
life, to put forth separately ; and the plan would probably have 
been carried into execution had riot his death intervened. It was 
translated into English, and published in 1736 by John Colson, 
Professor of Mathematics in Cambridge. 

Newton, it is thought, made his discoveries concerning the 
INFLECTION and DIFFRACTION of light before 1674. The phe 
nomena of the inflection of light had been first discovered more 
than ten years before by Grimaldi. And Newton began by re 
peating one of the experiments of the learned Jesuit admitting 
a beam of the sun s light through a small pin hole into a dark 
chamber : the light diverged from the aperture in the form of a, 
cone, and the shadows of all bodies placed in this light were 
larger than might have been expected, and surrounded with three 
coloured fringes, the nearest being widest, and the most remote 
the narrowest. Newton, advancing upon this experiment, took 
exact measures of the diameter of the shadow of a human hair, 
and of the breadth of the fringes, at different distances behind it, 
and discovered that these diameters and breadths were not pro 
portional to the distances at which they were measured. He 
hence supposed that the rays which passed by the edge of the 
hair were deflected or turned aside from it, as if by a repulsive 
force, the nearest rays suffering the greatest, the more remote a 
less degree of deflection. In explanation of the coloured fringes, 
he queried : whether the rays which differ in refrangibility do not 
differ also in flexibility, and whether they are n<t, by these dif 
ferent inflections, separated from one another, so as after separa- 



< LIFE OF SIR ISAAC NEWTON. 

tion to make the colours in the three fringes above described ? 
Also, whether the rays, in passing by the edges and sides ol 
bodies, are not bent several times backwards and forwards with 
an eel-like motion the three fringes arising from three such 
bendings ? His inquiries on this subject were here interrupted 
and fiever renewed. 

His Theory of the COLOURS of NATURAL BODIES was commu 
nicated to the Royal Society, in February, 1675. This is justly 
regarded as one of the profoundest of his speculations. The fun 
damental principles of the Theory in brief, are : That bodies 
possessing the greatest refractive powers reflect the greatest 
quantity of light ; and that, at the confines of equally refracting 
media, there is no reflection. That the minutest particles of al 
most all natural bodies are in some degree transparent. That 
between the particles of bodies there are pores, or spaces, either 
empty or filled with media of a less density than the particles 
themselves. That these particles, and pores or spaces, have some 
definite size. Hence he deduced the Transparency, Opacity, and 
colours of natural bodies. Transparency arises from the particles 
and their pores being too small to cause reflection at their com 
mon surfaces the light all passing through ; Opacity from the 
opposite cause of the particles and their pores being sufficiently 
large to reflect the light which is " stopped or stifled 7 by the 
multitude of reflections ; and colours from the particles, accord 
ing to their several sizes, reflecting rays of one colour and trans 
mitting those of another or in other words, the colour that 
meets the eye is the colour reflected, while all the other rays are 
transmitted or absorbed. 

Analogous in origin to the colours of natural bodies, he con 
sidered the COLOURS OF THIN PLATES. This subject was interest 
ing and important, and had attracted considerable investigation. 
He, however, was the first to determine the law of the produc 
tion of these colours, arid, during the same year made known the 
results of his researches herein to the Royal Society. His mode 
of procedure in these experiments was simple and curious. He 
placed a double convex lens of a large known radius of curvature, 
the flat surface of a plano-convex object glass. Thus, from 



UFE OF SIR ISAAC NEWTON. 25 

their point of contact at the centre, to the circumference of the 
lens, he obtained plates of air, or spaces varying from the ex- 
tremest possible thinness, by slow degrees, to a considerable thick 
ness. Letting the light fall, every different thickness of this 
plate of air gave different colours the point of contact of the 
lens and glass forming the centre of numerous concentric colored 
nags. Now the radius of curvature of the lens being known, the 
thickness of the plate of air, at any given point, or where any par 
ticular colour appeared, could be exactly determined. Carefully 
noting, therefore, the order in which the different colours ap 
peared, he measured, with the nicest accuracy, the different thick* 
nesses at which the most luminous parts of the rings were pro 
duced, whether the medium were air, water, or mica all these 
substances giving the same colours at different thicknesses ; the 
ratio of which he also ascertained. From the phenomena obser 
ved in these experiments, Newton deduced his Theory of Fits of 
EASY REFLECTION AND TRANSMISSION of light. It consists in suppos 
ing that every particle of light, from its first discharge from a lumi 
nous body, possesses, at equally distant intervals, dispositions to 
be reflected from, or transmitted through the surfaces of bodies 
upon which it may fall. For instance, if the rays are in a Fit of 
Easy Reflection, they are on reaching the surface, repelled, 
thrown off] or reflected from it ; if, in a Fit of Easy Transmission, 
they are attracted, drawn in, or transmitted through it. By this 
Theory of Fits, our author likewise explained the colours of 
thick plates. 

He regarded light as consisting of small material particles 
emitted from shining substances. He thought that these parti 
cles could be re-combined into solid matter, so that " gross bodies 
and light, were convertible into one another ;" that the particles of 
light and the particles of solid bodies acted mutually upon each 
other ; those of light agitating and heating those of solid bodies, 
and the latter attracting and repelling the former. Newton was 
the first to suggest the idea of the POLARIZATION of light. 

In the paper entitled An Hypothesis Explaining Properties of 
Light, December, 1675, our author first introduced his opinions re 
specting Ether opinions which he afterward abandoned and again 



26 LIFE OF SIR S.\AC 1SEWTON. 

permanently resumed " A most subtle spirit which pervades" ah 
bodies, and is expanded through all the heavens. It is electric, 
and almost, if not quite immeasurably elastic and rare. " By the 
force and action of which spirit the particles of bodies mutually 
attract one another, at near distances, and cohere, if contiguous ; 
and electric bodies operate at greater distances, as well repelling 
as attracting the neighbouring corpuscles ; and light is emitted, 
-reflected, refracted, inflected and heats bodies ; and all sensation 
is excited, and the members of animal bodies move at the com 
mand of the will, namely, by the vibrations of this spirit, mutu 
ally propagated along the solid filaments of the nerves, from the 
outward organs of sense to the brain, and from the brain into the 
muscles." This " spirit" was no anima mundi ; nothing further 
from the thought of Newton ; but was it not, on his part, a par 
tial recognition of, or attempt to reach an ultimate material force, 
or primary element, by means of which, " in the roaring loom of 
time," this material universe, God s visible garment, may be 
woven for us ? 

The Royal Society were greatly interested in the results of 
some experiments, which our author had, at the same time, com 
municated to them relative to the excitation of electricity in glass ; 
and they, after several attempts and further direction from him, 
succeeded in re-producing the same phenomena. 

One of the most curious of Newton s minor inquiries related to 
the connexion between the refractive powers and chemical com 
position of bodies. He found on comparing the refractive powers 
and the densities of many different substances, that the former 
were very nearly proportional to the latter, in the same bodies. 
Unctuous and sulphureous bodies were noticed as remarkable excep 
tions as well as the diamond their refractive powers being two 
or three times greater in respect of their densities than in the 
case of other substances, while, as among themselves, the one was 
generally proportional to the other. He hence inferred as to the 
diamond a great degree of combustibility ; a conjecture which 
the experiments of modern chemistry have shown to be true. 

The chemical researches of our author were probably pursued 
with more or less diligence from the time of his witnessing some 



LIFE OF .SIR ISAAC NEWTON. 27 

?t the uractical operations in that science at the Apothecary s at 
Grantham. DE NATURA ACIDORUM is a short chemical paper, on 
various topics, and published in Dr. Horsley s Edition of his 
works. TABULA QUANTITATUM E r GRADUUM COLORIS was in 
serted iii the Philosophical Transactions ; it contains a compara 
tive scale of temperature from that of melting ice to that of a 
small kitchen coal-fire. He regarded fire as a body heated so hot 
as to emit light copiously ; and flame as a vapour, fume, or ex 
halation heated so hot as to shine. To elective attraction, by 
the operation of which the small particles of bodies, as he con 
ceived, act upon one another, at distances so minute as to escape 
observation, he ascribed all the various chemical phenomena ot 
precipitation, combination, solution, and crystallization, and the 
mechanical phenomena of cohesion and capillary attraction. New 
ton s chemical views were illustrated and confirmed, in part, at 
least, in his own life-time. As to the structure of bodies, he was 
of opinion " that the smallest particles of matter may cohere by 
the strongest attractions, and compose bigger particles of weaker 
virtue ; and many of these may cohere and compose bigger par 
tides whose virtue is still weaker ; and so on for divers succes 
sions, until the progression end in the biggest particles, on which 
the operations in chemistry and the colours of natural bodies de 
pend, and which by adhering, compose bodies of sensible magni 
tude." 

There is good reason to suppose that our author was a diligent 
student of the writings of Jacob Behmen ; and that in conjunction 
with a relative, Dr. Newton, he was busily engaged, for several 
months in the earlier part of life, in quest of the philosopher s 
tincture. " Great Alchymist," however, very imperfectly de 
scribes the character of Behmen, whose researches into things 
material and things spiritual, things human and things divine, ai- 
ford the strongest evidence of a great and original mind. 

More appropriately here, perhaps, than elsewhere, may be 
given Newton s account of some curious experiments, made in his 
own person, on the action of light upon the retina, Locke, who 
was an intimate friend of our author, wrote to him for his opinion 
on a certain fact stated in Boyle s Book of Colours. Newton, in 



2S LIFE OF SIR ISAAC NEWTON. 

his reply, dated June 30th, 16 ( Jl, narrates the following circum 
stances, which probably took place in the course of his optical 
researches. Thus : 

" The observation you mention in Mr. Boyle s Book of Colours 
I once tried upon myself with the hazard of my eyes. The 
manner was this ; I looked a very little while upon the sun in the 
looking-glass with my right eye, and then turned my eyes into a 
dark corner of my chamber, arid winked, to observe the impres 
sion made, and the circles of colours which encompassed it, and 
how they decayed by degrees, and at last vanished. This I re 
peated a second and a third time. At the third time, when the 
phantasm of light and colours about it were almost vanished, in 
tending my fancy upon them to see their last appearance, I found, 
to my amazement, that they began to return, and by little and 
little to become as lively and vivid as when I had newly looked 
upon the sun. But when I ceased to intend my fancy upon them, 
they vanished again. After this, I found, that as often as I went 
into the dark, and intended my mind upon them, as when a man 
looks earnestly to see anything which is difficult to be seen, I 
could make the phantasm return without looking any more upon 
the sun ; and the oftener I made it return, the more easily I could 
make it return again. And, at length, by repeating this, without 
looking any more upon the sun, I made such an impression on my 
eye, that, if I looked upon the clouds, or a book, or any bright 
object, I saw upon it a round bright spot of light like the sun, 
and, which is still stranger, though I looked upon the sun with 
my right eye only, and not with my left, yet my fancy began *o 
make an impression upon my left eye, as well us upon my right. 
For if I shut my right eye, or looked upon a book, or the clouds, 
with my left eye, I could see the spectrum of the sun almost as 
plain as with my right eye, if I did but intend my fancy a little 
while upon it ; for at first, if I shut my right eye, and looked with 
my left, the spectrum of the sun did not appear till I intended my 
fancy upon it ; but by repeating, this appeared every time more 
easily. And now, in a few hours time, I had brought my eyes 
to such a pass, that I could look upon no blight object with either 
eye, but I saw the sun before me, so that I durst neither write 



LIFE OF SIR ISAAC NEWTON. 29 

nor read ; but to recover the use of my eyes, shut myself up in 
my chamber made dark, for three days together, and used all 
means to divert my imagination from the sun. For if I thought 
upon him, I presently saw his picture, though I was in the dark. 
But by keeping in the dark, and employing my mind about other 
things, I began in three or four days to have some use of my eyes 
again ; and by forbearing to look upon bright objects, recovered 
them pretty well, though not so well but that, for some months 
after, the spectrum of the sun began to return as often as I began 
to meditate upon the phenomena, even though I lay in bed at mid 
night with my curtains drawn. But now I have been very well 
for many years, though I am apt to think, if I durst venture my 
eyes, I could still make the phantasm return by the power of my 
fancy. This story I tell you, to let you understand, thaj; in the 
observation related by Mr. Boyle, the man s fancy probably con 
curred with the impression made by the sun s light to produce 
that phantasm of the sun which he constantly saw in bright ob 
jects. And so your question about the cause of phantasm in 
volves another about the power of fancy, which I must confess is 
too hard a knot for me to untie. To place this effect in a constant 
motion is hard, because the sun ought then to appear perpetually. 
It seems rather to consist in a disposition of the sensorium to 
move the imagination strongly, and to be easily moved, both by 
the imagination and by the light, as often as bright objects are 
looked upon." J 

Though Newton had continued silent, yet his thoughts were 
by no means inactive upon the vast subject of the planetary mo 
tions. The idea of Universal Gravitation, first caught sight of, so 
to speak, in the garden at Woolsthorpe, years ago, had gradually 
expanded upon him. We find him, in a letter to Dr. Hooke, 
Secretary of the Royal Society, dated in November, 1679, pro 
posing to verify the motion of the earth by direct experiment, 
namely, by the observation of the path pursued by a body falling 
from a considerable height. He had concluded that the path 
would be spiral ; but Dr. Hooke maintained that it would be an 
eccentric ellipse iu vacuo, and an ellipti-spiral in a resisting me 
dium. Our author, aided by this correction of his error, and by 



30 LIFE OF SIR ISAAC NEWTON. 

the discovery that a projectile would move in an elliptical orbil 
when under the influence of a force varying inversely as the 
square of the distance, was led to discover " the theorem bj 
which he afterwards examined the ellipsis ;" and to demonstrate 
the celebrated proposition that a planet acted upon by an attrac 
tive force varying inversely as the squares of the distances will 
describe an elliptical orbit, in one of whose foci the attractive 
force resides. 

When he was attending a meeting of the Royal Society, in 
June 1682, the conversation fell upon the subject of the measure 
ment of a degree of the meridian, executed by M. Picard, a 
French Astronomer, in 1679. Newton took a memorandum oi 
the result ; and afterward, at the earliest opportunity, computed 
from it the diameter of the earth : furnished with these new data, 
he resumed his calculation of 1666. As he proceeded therein, 
he saw that his early expectations were now likely to be realized ; 
the thick rushing, stupendous results overpowered him ; he be 
came unable to carry on the process of calculation, and intrusted 
its completion to one of his friends. The discoverer had, indeed, 
grasped the master-fact. The law of falling bodies at the earth s 
surface was at length identified with that which guided the moon 
in her orbit. And so his GREAT THOUGHT, that had for sixteen 
years loomed up in dim, gigantic outline, amid the first dawn of a 
plausible hypothesis, now stood forth, radiant and not less grand, 
in the mid-day light of demonstrated truth. 

It were difficult, nay impossible to imagine, even, the influence 
of a result like this upon a mind like Newton s. It was as if the 
keystone had been fitted to the glorious arch by which his spirit 
should ascend to the outskirts of infinite space spanning the immea 
surable weighing the imponderable computing the incalculable 
mapping out the marchings of the planets, and the far-wander 
ings of the come f s, and catching, bring back to earth some clearer 
notes of that higher melody which, as a sounding voice, bears 
perpetual witness to the design and omnipotence of a creating 
Deity. 

Newton, extending the law thus obtained, composed a series 
of about twelve propositions on the motion of the primary planets 



LIFE OF SIR ISAAC NEWTON. 31 

about the sun. These were sent to London, and communicated 
to the Royal Society about the end of 1683. At or near this pe 
riod, other philosophers, as Sir Christopher Wren, Dr. Halley, 
and Dr. Hooke, were engaged in investigating the same subject ; 
but with no definite or satisfactory results. Dr. Halley, having 
seen, it is presumed, our author s propositions, went in August, 
1684, to Cambridge to consult with him upon the subject. 
Newton assured him that he had brought the demonstration to 
perfection. In November, Dr. Halley received a copy of the 
work ; and, in the following month^ announced . it to the Royal 
Society, with the author s promise to have it entered upon their 
Register. Newton, subsequently reminded by the Society of his 
promise, proceeded in the diligent preparation of the work, and. 
though suffering an interruption of six weeks, transmitted the 
manuscript of the first book to London before the end of April. 
The work was entitled PHILOSOPHI/E NATURALIS PRINCIPIA 
MATHEMATICA, dedicated to the Royal Society, and presented 
thereto on the 28th of April, 1685-6. The highest encomiums 
were passed upon it ; and the council resolved, on the 19th of 
May, to print it at the expense of the Society, and under the di 
rection of Dr. Halley. The latter, a few days afterward, com 
municated these steps to Newton, who, in a reply, dated the 20th 
of June, holds the following language : " The proof you sent me 
I like very well. I designed the whole to consist of three books ; 
the second was finished last summer, being short, and only wants 
transcribing, and drawing the cuts fairly. Some new propositions 
I have since thought on, which I can as well let alone. The 
third wants the theory of comets. In autumn last, I spent two 
months in calculation to no purpose for want of a good method, 
which made me afterward return to the first book, and enlarge it 
with diverse propositions, some* relating to comets, others to other 
things found ou f last winter. The third I now design to sup 
press. Philosophy is such an impertinently litigious lady, that a 
man had as good be engaged in liw-suits as have to do with her. 
I found it so formerly, and now I can no sooner come near her 
again, but she gives me warning. The first two books without 
the third will not so well bear the title of P/iilosophicc Naturalis 

3 



32 LIFE OF SIR ISAAC NEWTON. 

Principia Mathematicia ; and thereupon I had altered it to this, 
De Motu Corporum Libri duo. But after second thought I re 
tain the former title. It will help the sale of the book, which I 
ought not to diminish now tis yours." 

This " warning" arose from some pretensions put forth by Dr. 
Hooke. And though Newton gave a minute and positive refuta 
tions of such claims, yet, to reconcile all differences, he gener 
ously added to Prop. IV. Cor. 6, Book I, a Scholium, in which 
Wren, Hooke and Halley are acknowledged to have indepen 
dently deduced the law of gravity from the second law of 
Kepler. 

The suppression of the third book Dr. Halley could not endure 
to see. " I must again beg you" says he, " not to let your re 
sentments run so high as to deprive us of your third book, where 
in your applications of your mathematical doctrine to the theory 
of comets, and several curious experiments, which, as I guess by 
what you write ought to compose it, will undoubtedly render it 
acceptable to those who will call themselves philosophers without 
mathematics, which are much the greater number." To these 
solicitations Newton yielded. There were no "resentments," how 
ever, as we conceive, in his " design to suppress." He sought 
peace ; for he loved and valued it above all applause. But, in 
spite of his efforts for tranquillity s sake, his course of discovery 
was all along molested by ignorance or presumptuous rivalry. 

The publication of the great work now went rapidly forwards, 
The second book was sent to the Society, and presented on the 
2d March ; the third, on the 6th April ; and the whole was com 
pleted and published in the month of May, 1686-7. In the sec 
ond Lemma of the second book, the fundamental principle of his 
fiuxionary calculus was, for the first time, given to the world ; but 
its algorithm or notation did not appear till published in the 
second volume nf Dr. Wallis s works, in 1693. 

And thus was ushered into existence The PRINCIPIA a work 
to which pre-eminence above all the productions of the human 
intellect has been awarded a work that must be esteemed of 
priceless worth so long as Science has a votary, or a single wor 
shipper be left to kneel at the altar of Truth. 



LIFE OF SIR ISAAC NEWTON. 33 

The entire work bears the general title of THE MATHEMATICAL 
PRINCIPLES OF NATURAL PHILOSOPHY. It consists of three books: 
the first two, entitled, OF THE MOTION OF BODIES, are occupied 
with the laws and conditions of motions and forces, and are illus 
trated with many scholia treating of some of the most general 
and best established points in philosophy, such as the density and 
resistance of bodies, spaces void of matter, and the motion of 
sound and light. From these principles, there is deduced, in the 
third book, drawn up in as popular a style as possible and entitled, 
OF THE SYSTEM OF THE WORLD, the constitution of the system of 
i he world. In regard to this book, the author say^ " I had, indeed, 
composed the third Book in a popular method, that it might be read 
by many ; but afterwards, considering that such as had not suf- 
ficently entered into the principles could not easily discover the 
strength of the consequences, nor lay aside the prejudices to which 
they had been many years accustomed, therefore, to prevent dis 
putes which might be raised upon such accounts, I chose to reduce 
the substance of this Book into the form of Propositions (in the 
mathematical way), which should be read by those only who had 
first made themselves masters of the principles established in the 
preceding Books : not that I would advise any one to the previous 
study of every Proposition of those Books." "It is enough it 
one carefully reads the Definitions, the Laws of Motion, and the 
three first Sections of the first Book. He may then pass on to 
this Book, and consult such of the remaining Propositions of the 
first two Books, as the references in this, and his occasions shall re 
quire." So that " The System of the World" is composed both 
" in a popular method," and in the form of mathematical Propo 
sitions. 

The principle of Universal Gravi ition, namely, that every 
particle of matter is attracted by, or gravitates to, every other 
particle of matter, icith a force inversely proportional to the 
squares of their distances is the discovery w? ich characterizes 
The PRINCIPIA. This principle the author deduced from the mo 
tion of the moon, and the three laws of Kepler laws, which 
Newton, in turn, by his greater law, demonstrated to be true. 

From the first law of Kepler, namely, the proportionality of 



LIFE OF SIR ISAAC NEWTON. 

the areas to t\ie times of their description, our author inferred 
that the force which retained the planet in its orbit was always 
directed to the sun ; and from the second, namely, that every 
planet moves in an ellipse with the sun in one of its foci, he drew 
the more general inference that the force by which the planet 
moves round that focus varies inversely as the square of its dis 
tance therefrom : and he demonstrated that a planet acted upon 
by such a force could not move in any other curve than a conic 
section ; showing when the moving body would describe a circu 
lar, an elliptical, a parabolic, or hyperbolic orbit. He demon 
strated, too, that this force, or attracting, gravitating power re 
sided in every, the least particle ; but that, in spherical masses, it 
operated as if confined to their centres ; so that, one sphere or 
body will act upon another sphere or body, with a force directly 
proportional to the quantity of matter, and inversely as the square 
of the distance between their centres; and that their velocities of 
mutual approach will be in the inverse ratio of their quantities o* 
matter. Thus he grandly outlined the Universal Law. Verify 
ing its truth by the motions of terrestrial bodies, then by those of 
the moon and other secondary orbs, he finally embraced, in one 
mighty generalization, the entire Solar System all the move 
ments of all its bodies planets, satellites and comets explain 
ing and harmonizing the many diverse and theretofore inexplica 
ble phenomena. 

Guided by the genius of Newton, we see sphere bound to 
sphere, body to body, particle to particle, atom to mass, the min 
utest part to the stupendous whole each to each, each to all, 
and all to each in the mysterious bonds of a ceaseless, recipro 
cal influence. An influence whose workings are shown to be 
alike present in the globular dew-drop, or oblate-spheroidal earth ; 
in the falling shower, or vast heaving ocean tides ; in the flying 
thistle-down, or fixed, ponderous rock ; in the swinging pendulum, 
or time-measuring sun ; in the varying and unequal moon, or 
earth s slowly retrograding poles ; in the uncertain meteor, or 
oiazing comet wheeling swiftly away on its remote, yet determined 
round. An influence, in fine, that may link system to system 
through all the star-glowing firmament ; then firmament to iirma- 



LIFE OF SIR ISAAC NEWTON. 35 

merit ; aye, firmament to firmament, again and again, till, con 
verging home, it may be, to some ineffable centre, where more 
presently dwells He who inhabiteth immensity, and where infini 
tudes meet and eternities have their condux, and where around 
move, in softest, swiftest measure, all the countless hosts that 
crowd heaven s fathomless deeps. 

And yet Newton, amid the loveliness and magnitude of Om 
nipotence, lost not sight of the Almighty One. A secondary, 
however universal, was not taken for the First Cause. An im 
pressed force, however diffused and powerful, assumed not the 
functions of the creating, giving Energy. Material beauties, 
splendours, and sublimities, however rich in glory, and endless in 
extent, concealed not the attributes of an intelligent Supreme. 
From the depths of his own soul, through reason and the WORD, 
he had risen, a priori, to God : from the heights of Omnipotence, 
through the design and law of the builded universe, he proved </ 
posteriori, a Deity. " I had," says he, " an eye upon such prin 
ciples as might work, with considering men, for the belief of a 
Deity," in writing the PRINCIPIA ; at the conclusion whereof, he 
teaches that " this most beautiful system of the sun, planets and 
comets, could only proceed from the counsel and dominion of an 
intelligent and powerful Being. And if the fixed stars are the 
centres of other like systems, these, being forme 1 by the like 
wise counsels, must be all subject to the dominion of One ; especially 
since the light of the fixed stars is of the same nature with the 
light of the sun, and from every system light passes into all other 
systems : and lest the systems of the fixed stars should, by their 
gravity, fall on each other mutually, he hath placed those systems 
at immense distances one from another. 

" This Being governs all things, not as the soul of the world, 
but as Lord over all ; and on account of his dominion he is wont, 
to be called Lord God Travrowparwp or Universal Ruler ; for God 
is a relative word, and has a respect to servants ; and Deity is 
the dominion of God, not over his own body, as those imagine 
who fancy God to be the soul of the world, but over servants. 
The Supreme God is a Being eternal, infinite, absolutely perfect ; 
but a being, however perfect, without dominion, cannot be said to 



36 LIFE OF SIR ISAAC NEWTON. 

be Lord God ; for we say, my God, your God, the God of Israel 
the God of Gods, and Lord of Lords ; but we do not say, my 
Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods : 
we do not say my Infinite, or my Perfect : these are titles which 
have no respect to servants. The word God usually signifies 
Lord ; but every Lord is not God. It is the dominion of a spir 
itual Being which constitutes a God ; a true, supreme, or imagi 
nary dominion makes a true, supreme, or imaginary God. And 
from his true dominion it follows that the true God is a living, 
intelligent and powerful Being ; and from his other perfections, 
that he is supreme or most perfect. He is eternal and in 
finite, omnipotent and omniscient ; that is, his duration reaches 
from eternity to eternity ; his presence from infinity to infinity ; 
he governs all things and knows all things, that are or can be 
done. He is not eternity or infinity, but eternal and infinite ; 
he is not duration and space, but he endures and is present. 
He endures forever and is everywhere present ; and by existing 
always and everywhere, he constitutes duration and space. Since 
every particle of space is always, and every indivisible moment 
of duration is everywhere, certainly the Maker and Lord of things 
cannot be never and nowhere. Every soul that has perception 
is, though in different times and different organs of sense and mo 
tion, still the same indivisible person. There are given succes 
sive parts in duration, co-existent parts in space, but neither the 
one nor the other in the person of a man, or his thinking 
principle ; and much less can they be found in the thinking sub 
stance of God. Every man. so far as he is a thing that has j:er- 
ceptiori, is one and the same man during his whole life, in all and 
each of his organs of sense. God is one and the same God, al 
ways and everywhere. He is omnipresent, not virtually only, 
but also substantially ; for virtue cannot subsist without sub 
stance. In him are all things contained and moved ; yet neither 
affects the other ; God suffers nothing from the motion of bodies ; 
bodies find no resistance from the omnipresence of God. It is 
allowed by all that the Supreme God exists necessarily ; and by 
the same necessity he exists always and everywhere. Whence 
also he is all similar, all eye, all ear, all brain, all arm, all powei 



LIFE CF SIR ISAAC NEWTON. 37 

to perceive, to understand, and to act ; but in a manner not at all 
human, in a manner not at all corporeal, in a manner utterly un 
known to us. As a blind man has no idea of colours, so have we 
no idea of the manner by which the all-wise God perceives and 
understands all things. He is utterly void of all body, and bodily 
figure, and can therefore neither be seen, nor heard, nor touched ; 
nor ought he to be worshipped under the representation of any 
corporeal thing. We have ideas of his attributes, but what the 
real substance of anything is we know not. In bodies we see 
only their figures and colours, we hear only the sounds, we touch 
only their outward surfaces, we smell only the smells, and taste 
only the savours ; but their inward substances are not to be known, 
either by our senses, or by any reflex act of our minds : much 
less, then, have we any idea of the substance of God. We know 
him only by his most wise and excellent contrivances of things, 
and final causes ; we admire him for his perfections ; but we rev 
erence and adore him on account of his dominion ; for we adore 
him as his servants ; and a god without dominion, providence, and 
final causes, is nothing else but Fate and Nature. Blind meta 
physical necessity, which is certainly the same always and every 
where, could produce no variety of things. All that diversity of 
natural things which we find suited to different times and places 
could arise from nothing but the ideas and will of a Being neces 
sarily existing." 

Thus, the diligent student of science, the earnest seeker of 
truth, led, as through the courts of a sacred Temple, wherein, at 
each step, new wonders meet the eye, till, as a crowning grace, 
they stand before a Holy of Holies, and learn that all science and 
all truth are one which hath its beginning and its end in the 
knowledge of Him whose glory the heavens declare, and whose 
handiwork the firmament showeth forth. 

The introduction of the pure and lofty doctrines of the PRIN- 
CIPIA was perseveringly resisted. Descartes, with his system of 
vortices, had sown plausibly to the imagination, and error had 
struck down deeply, and shot up luxuriantly, not only in the 
popular, but in the scientific mind. Besides the idea in itself so 
simple and so grand that the great masses of the planets were 



38 LIFE OF SIR ISAAC NEWTON. 

suspended in empty space, and retained in their orbits by an in 
visible influence residing in the sun was to the ignorant a thing 
inconceivable, and to the learned a revival of the occult qualities 
of the ancient physics. This remark applies particularly to the 
continent. Leibnitz misapprehended ; Huygens in part rejected ; 
John Bernouilli opposed ; and Fontenelle never received the doc 
trines of the PRINCIPIA. So that, the saying of Voltaire is prob 
ably true, that though Newton survived the publication of his 
great work more than forty years, yet, at the time of his death, 
lie had not above twenty followers out of England. 

But in England, the reception of our author s philosophy was 
rapid and triumphant. His own labours, while Lucasian Pro 
fessor ; those of his successors in that Chair Whiston and 
Saunderson ; those of Dr. Samuel Clarke, Dr. Laughton, Roger 
Cotes, and Dr. Bentley ; the experimental lectures of Dr. Keill 
and Desaguliers ; the early and powerful exertions of David 
Gregory at Edinburgh, and of his brother James Gregory at St. 
Andrew s, tended to diffuse widely in England and Scotland a 
knowledge of, and taste for the truths of the PRINCIPIA. Indeed, 
its mathematical doctrines constituted, from the first, a regular 
part of academical instruction ; while its physical truths, given to 
the public in popular lectures, illustrated by experiments, had, 
before the lapse of twenty ) ( ar.s, become familiar to, and adopted 
by the general mind. Pemberton s popular " View of Sir Isaac 
Newton s Philosophy" was published, in 1728 ; and the year after 
ward, an English translation of the PRINCIPIA, and System of the 
World, by Andrew Motte. And since that period, the labours of 
Le Seur and Jacquier, of Thorpe, of Jebb, of Wright and others 
have greatly contributed to display the most hidden treasures of 
the PRINCIPIA. 

About the time of the publication of the Principia, James II., 
bent on re-establishing the Romish Faith, had, among other ille 
gal acts, ordered by mandamus, the University of Cambridge to 
confer the degree of Master of Arts upon an ignorant monk. 
Obedience to this mandate was resolutely refused. Newton was 
one of the nine delegates chosen to defend the independence of 
the University. They appeared before the High Court ; and 



LIFE OF SIR ISAAC NEWTON. 39 

successfully : the king abandoned his design. The prominent 
part which our author took in these proceedings, and his eminence 
in the scientific world, induced his proposal as one of the parlia 
mentary representatives of the University. He was elected, in 
1688, and sat in the Convention Parliament till its dissolution. 
After the first year, however, he seems to have given little or no 
attention to his parliamentary duties, being seldom absent from 
the University till his appointment in the Mint, in 1695. 

Newton began his theological researches sometime previous to 
1691 ; in the prime of his years, and in the matured vigour of 
his intellectual powers. From his youth, as we have seen, he 
had devoted himself with an activity the most unceasing, and an 
energy almost superhuman to the discovery of physical truth ; 
giving to Philosophy a new foundation, and to Science a new 
temple. To pass on, then, from the consideration of the material, 
more directly to that of the spiritual, was a natural, nay, with so 
large and devout a soul, a necessary advance. The Bible was to 
him of inestimable worth. In the elastic freedom, which a pure 
and unswerving faith in Him of Nazareth gives, his mighty facul 
ties enjoyed the only completest scope for development. His 
original endowment, however great, combined with a studious 
application, however profound, would never, without this libera 
tion from the dominion of passion and sense, have enabled him to 
attain to that wondrous concentration and grasp of intellect, for 
which Fame has as yet assigned him no equal. Gratefully he 
owned, therefore, the same Author in the Book of Nature and the 
Book of Revelation. These were to him as drops of the same 
unfathomable ocean ; as outrayings of the same inner splendour ; 
as tones of the same ineffable voice ; as segments of the same 
infinite curve. "With great joy he had found himself enabled to 
proclaim, as an interpreter, from the hieroglyphs of Creation, the 
existence of a God : and now, with greater joy, and in the fulness 
of his knowledge, and in the fulness of his strength, he laboured 
to make clear, from the utterances of the inspired Word, the far 
mightier confirmations of a Supreme Good, in all its glorious 
amplitude of Being and of Attribute ; and to bring the infallible 
workings thereof plainly home to the understandings and the 



40 LIFE OF SIR ISAAC NEWTON. 

affections of his fellow-men ; and finally to add the weight of his 
own testimony in favour of that Religion, whose truth is now. in 
deed, " girded with the iron and the rock of a ponderous and co 
lossal demonstration." 

His work, entitled, OBSERVATIONS UPON THE PROPHECIES OF 
HOLY WRIT, PARTICULARLY THE PROPHECIES OF DANIEL AND THE 
APOCALYPSE OF ST. JOHN, first published in London, in 1733 4to. 
consists of two parts : the one devoted to the Prophecies oi 
Daniel, and the other to the Apocalypse of St. John. In the first 
part, he treats concerning the compilers of the books of the Old 
Testament ; of the prophetic language ; of the vision of the 
four beasts ; of the kingdoms represented by the feet of the 
image composed of iron and clay ; of the ten kingdoms repre 
sented by the ten horns of the beast ; of the eleventh horn of 
Daniel s fourth beast ; of the power which should change times 
and laws ; of the kingdoms represented in Daniel by the ram 
and he-goat ; of the prophecy of the seventy weeks ; of the 
times of the birth and passion of Christ ; of the prophecy of the 
Scripture of Truth ; of the king who doeth according to his will, 
and magnified himself above every god, and honoured Mahuzzims, 
and regarded not the desire of women ; of the Mahuzzim, hon 
oured by the king who doeth according to his will. In the sec 
ond part, he treats of the time when the Apocalypse was written , 
of the scene of the vision, and the relation which the Apocalypse 
has to the book of the law of Moses, and to the worship of God 
in the temple ; of the relation which the Apocalypse has to the 
prophecies of Daniel, and of the subject of the prophecy itself 
Newton regards the prophecies as given, not for the gratification 
of man s curiosity, by enabling him to foreknow ; but for his con 
viction that the world is governed by Providence, by witnessing 
their fulfilment. Enough of prophecy, he thinks, has already 
been fulfilled to afford the diligent seeker abundant evidence of 
God s providence. The whole work is marked by profound 
erudition, sagacity and argument. 

And not less learning, penetration and masterly reasoning are 
conspicuous in his HISTORICAL ACCOUNT OF Two NOTABLE 
CORRUPTIONS OF SCRIPTURES IN A LETTER TO A FRIEND. This 



LIFE OF SIR ISAAC NEWTON. 41 

Treatise, first accurately published in Dr. Horsley s edition of his 
works, relates to two texts : the one, 1 Epistle of St. John v. 7 ; 
the other, 1 Epistle of St. Paul to Timothy iii. 16. As this 
work had the effect to deprive the advocates of the doctrine of 
the Trinity of two leading texts, Newton has been looked upon 
as an Arian ; but there is absolutely nothing in his writings to 
warrant such a conclusion. 

His regaining theological works consist of the LEXICON PRO- 
PHETICUM, which was left incomplete ; a Latin Dissertation on 
the sacred cubit of the Jews, which was translated into English, 
and published, in 1737. among the Miscellaneous Works of John 
Greaves ; and FOUR LETTERS addressed to Dr. Bentlty, contain 
ing some arguments in proof of a Deity. These Letters were 
dated respectively : 10th December, 1692 ; 17th January, 1693 ; 
25th February, 1693; and llth February, 1693 the fourth 
bearing an earlier date than the third. The best faculties and 
the profoundest acquirements of our author are convincingly 
manifest in these lucid and powerful compositions. They were 
published in 1756, and reviewed by Dr. Samuel Johnson. 

Newton s religious writings are distinguished by their absolute 
freedom from prejudice. Everywhere, throughout them, there 
glows the genuine nobleness of soul. To his whole life, indeed, 
we may here fitly extend the same observation. He was most 
richly imbued with the very spirit of the Scriptures which he so 
delighted to study and to meditate upon. His was a piety, so 
fervent, so sincere and practical, that it rose up like a holy incense 
from every thought and act. His a benevolence that not only 
willed, but endeavoured the best for all. His a philanthropy 
that held in the embracings of its love every brother-man. 
His a toleration of the largest and the truest ; condemning per 
secution in every, even its mildest form ; and kindly encouraging 
each striving after excellence : .1 toleration that came not of 
indifference for the immoral and the impious met with their 
quick rebuke but a toleration that came of the wise humbleness 
and the Christian charity, which see, in the nothingness of self 
and the almightiness of TRUTH, no praise for the ablest, and no 
blame for th^ feeblest in their strugglings upward to light and life. 



42 LIFE OF SIR ISAAC NEWTON, 

Tn the winter of 1691-2, on returning from chapel, one morn 
ing, Newton foima tnat a favourite little dog, called Diamond, 
had overturned a lighted taper on his desk, and that several pa 
pers containing the results of certain optical experiments, were 
nearly consumed. His only exclamation, on perceiving his loss, 
was, " Oh Diamond, Diamond, little knowest thou the mischiel 
thou hast done," Dr. Brewster, in his life of our author, gives the 
following extract from the manuscript Diary of Mr. Abraham De 
La Pryme. a student in the University at the time of this oc 
currence. 

" 1692. February, 3. What I heard to-day I must relate. 
There is one Mr. Newton (whom I have very oft seen), Fellow 
of Trinity College, that is mighty famous for his learning, being a 
most excellent mathematician, philosopher, divine, &c. He has 
been Fellow of the Royal Society these many years ; and among 
other very learned books and tracts, he : s written one upon the mathe 
matical principles of philosophy, which has given him a mighty 
name, he having received, especially from Scotland, abundance of 
congratulatory letters for the same ; but of all the books he ever 
wrote, there was one of colours and light, established upon thou 
sands of experiments which he had been twenty years of making, 
and which had cost him many hundreds of pounds. This book 
which he vaiued so much, and which was so much talked of, had 
the ill luck to perish, and be utterly lost just when the learned 
author was almost at pitting a conclusion at the same, after this 
manner : In a winter s morning, leaving it among his other papers 
on his study table while he went to chapel, the candle, which he 
had unfortunately left burning there, too, catched hold by some 
means of other papers, and they fired the aforesaid book, and ut 
terly consumed it and several other valuable writings ; arid which 
is most wonderful did no further mischief. But when Mr. New 
ton came from chapel, and had seen what was done, every one 
thought he would have run mad, he was so troubled thereat that 
he was not himself for a month after. A long account of this his 
system of colours you may find in the Transactions of the Royal 
Society, which he had sent up to them long before this sad mis 
chance happened unto him." 



LIFE OF SIR ISAAC NEWTON. 43 

It will be borne in mind that all of Newton s theological wri 
tings, with the exception of the Letters to Dr. Bentley, were 
composed before this event which, we must conclude, from 
Pryme s words, produced a serious impression upon our author for 
about a month. But M. Biot, in his Life of Newton, relying on a 
memorandum contained in a small manuscript Journal of Huygens, 
declares this occurrence to have caused a deran-gement of New 
ton s intellect. M. Blot s opinions and deductions, however, as 
well as those of La Place, upon this subject, were based upon 
erroneous data, and have been overthrown by the clearest proof. 
There is not, in fact, the least evidence that Newton s reason was, 
for a single moment, dethroned ; on the contrary, the testimony 
is conclusive that he was, at all times, perfectly capable of carry 
ing on his mathematical, metaphysical and astronomical inquiries. 
Loss of sleep, loss of appetite, and irritated nerves will disturb 
somewhat the equanimity of the most serene ; and an act done, or 
language employed, under such temporary discomposure, is not a 
just criterion of the general tone and strength of a man s mind. 
As to the accident itself, we may suppose, whatever might have 
been its precise nature, that it greatly distressed him, and, still 
further, that its shock may have originated the train of nervous 
derangements, which afflicted him, more or less, for two years 
afterward. Yet, during this very period of ill health, we find him 
putting forth his highest powers. In 1692, he prepared for, and 
transmitted to Dr. Wallis the first proposition of the Treatise on 
Quadratures, with examples of it in first, second and third flux 
ions. He investigated, in the same year, the subject of haloes ; 
making and recording numerous and important observations rela 
tive thereto. Those profound and beautiful Letters to Dr. Bentley 
were written at the close of this and the beginning of the next 
year. In October, 1693, Locke, who was then about publishing a 
second edition of his work on the Human Understanding, request 
ed Newton to reconsider his opinions on innate ideas. And in 
1694, he was zealously occupied in perfecting his lunar theory ; 
visiting Flamstead, at the Royal Observatory of Greenwich, in 
September, and obtaining a series of lunar observations ; and 



14 LIFE OF SIR ISAAC NEWTON. 

commencing, in October, a correspondence with that distinguished 
practical Astronomer, which continued till 1698. 

We now arrive at the period when Newton permanently with 
drew from the seclusion of a collegiate, and entered upon a more 
active and public life. He was appointed Warden of the Mint, 
in 1695, through the influence of Charles Montague, Chancellor 
of the Exchequer, and afterward Earl of Halifax. The current 
roin of the nation had been adulterated and debased, and Mon 
tague undertook a re-coinage. Our author s mathematical and 
chemical knowledge proved eminently useful in accomplishing 
this difficult and most salutary reform. In 1699, he was pro 
moted to the Mastership of the Mint an office worth twelve or 
fifteen hundred pounds per annum, and which he held during the 
remainder of his life. He wrote, in this capacity, an official Re 
port on the Coinage, which has been published ; he also prepared 
a Table of Assays of Foreign Coins, which was printed at the 
end of Dr. Arbuthnot s Tables of Ancient Coins, Weights, and 
Measures, in 1727. 

Newton retained his Professorship at Cambridge till 1703. 
But he had, on receiving the appointment of Master of the Mint, 
in 1699, made Mr. Whiston his deputy, with all the emoluments 
of the office ; and, on finally resigning, procured his nomination to 
the vacant Chair. 

In January 1697, John Bernouilli proposed to the most distin 
guished mathematicians of Europe two problems for solution. 
Leibnitz, admiring the beauty of one of them, requested the time 
for solving it to be extended to twelve months twice the period 
originally named. The delay was readily granted. Newton, how 
ever, sent in, the day after he received the problems, a solution of 
them to the President of the Royal Society. Bernouilli obtained 
solutions from Newton, Leibinitz and the Marquis De L Hopital ; 
but Newton s though anonymous, he immediately recognised 
" tanquam ungue leonem" as the lion is known by his claw. 
We may mention here the famous problem of the trajectories 
proposed by Leibnitz, in 1716, for the purpose of "feeling the 
pulse of the English Analysts." Newton received the problem 
about five o clock in the afternoon, as he was returning from the 



LIFE OF SIR ISAAC NEWTON. 45 

Mint ; and though it was extremely difficult and he himself much 
fatigued, yet he completed its solution, the same evening before 
he went to bed. 

The history of these problems affords, by direct comparison, a 
striking illustration of Newton s vast superiority of mind. That 
amazing concentration and grasp of intellect, of which we have 
spoken, enabled him to master speedily, and, as it were, by a 
single effort, those things, for the achievement of which, the many 
would essay utterly in vain, and the very, very few attain only 
after long and renewed striving. And yet, with a modesty as 
unparalleled as his power, he attributed his successes, not to any 
extraordinary sagacity, but solely to industry and patient thought. 
Mr- kept the subject of consideration constantly before him, and 
waited till the first dawning opened gradually into a full and 
clear light ; never quitting, if possible, the mental process till the 
object of it were wholly gained. He never allowed this habit of 
meditation to appear in his intercourse with society ; but in the 
privacy of his own chamber, or in the midst of his own family, he 
gave himself up to the deepest abstraction. Occupied with some 
interesting investigation, he would often sit down on his bedside, 
after he rose, and remain there, for hours, partially dressed. 
Meal-time would frequently come and pass unheeded ; so that, 
unless urgently reminded, he would neglect to take the re 
quisite quantity of nourishment. But notwithstanding his anx 
iety to be left undisturbed, he would, when occasion required, 
turn aside his thoughts, though bent upon the most intricate re 
search, and then, when leisure served, again direct them to the 
very point where they ceased to act : and this he seemed to ac 
complish not so much by the force of his memory, as by the force 
of his inventive faculty, before the vigorous intensity of which, no 
subject, however abstruse, remained long unexplored. 

Me was elected a member of the Royal Academy of Sciences 
at Paris, in 1699, when that distinguished Body were empowered, 
by a new charter, to admit a small number of foreign associates. 
In 1700, he communicated to Dr. Halley a description of his re 
flecting instrument for observing the moon s distance from the 
fixed stars. This description was published in the Philosophical 



46 LIFE OF SIR ISAAC NEWTON, 

Transactions, in 1742. The instrument was the same as that 
produced by Mr. Hadley, in 1731, and which, under the name of 
Hadley s Quadrant, has been of so great use in navigation. On 
the assembling of the new Parliament, in 1701, Newton was re- 
elected one of the members for the University of Cambridge. In 
1703, he was chosen President of the Royal Society of London, 
to which office he was annually re-elected till the period of his 
decease about twenty-five years afterward. 

Our author unquestionably devoted more labour to, and, in 
many respects, took a greater pride in his Optical, than his other 
discoveries. This science he had placed on a new and indestruc 
tible basis ; and he wished not only to build, but to perfect the 
costly and glowing structure. He had communicated, before the 
publication of the PRINCIPIA, his most important researches on 
light to the Royal Society, in detached papers which were inserted 
in successive numbers of the Transactions ; but he did not pub 
lish a connected view of these labours till 1704, when they appeared 
under the title of OPTICS : OR, A TREATISE ON THE REFLEXIONS, 
REFRACTIONS, INFLEXIONS AND COLOURS OF LIGHT. To this, 
but to no subsequent edition, were added two Mathematical Trea 
tises, entitled, TRACTATUS DUO DE SPECIEBUS ET MAGNITUDINE 
FIGURARUM cuRViLiNEARUM ; the one bearing the title TRACTATUS 
DE QUADRATURA CuRVARUM ; and the other, that of ENUMERATIO 
LINEARUM TERTII ORDiNis. The publication of these Mathemati 
cal Treatises was made necessary in consequence of plagiarisms 
from the manuscripts of them loaned by the author to his friends. 
Dr. Samuel Clarke published a Latin translation of the Optics, in 
in 1706 ; whereupon he was presented by Newton, as a mark of 
his grateful approbation, with five hundred pounds, or one hun 
dred pounds for each of his children. The work was afterward 
translated into French. It had a remarkably wide circulation, 
and appeared, in several successive editions, both in England and 
on the Continent. There is displayed, particularly on this Opti 
cal Treatise, the author s talent for simplifying and communica 
ting the profoundest speculations. It is a faculty rarely united to 
that of the highest invention. Newton possessed both ; and thus 
that mental perfectness which enabled him to create, to combine, 



LIFE OF SIR ISAAC NEWTON. 47 

and to teach, and so render himself, not the "ornament" cnly ; 
but inconceivably more, the pre-eminent benefactor of his species. 

The honour of knighthood v/as conferred on our author in 
1705. Soon afterward, he was a candidate again for the Repre 
sentation of the University, but was defeated by a large majority. 
It is thought that a more pliant man was preferred by both min 
isters and electors. Newton was always remarkable for simplicity 
of dress, and his only known departure from it was on this oc 
casion, when he is said to have appeared in a suit of laced 
clothes. 

The Algebraical Lectures which he had, Juring nine years, 
delivered at Cambridge, were published by Whiston, in 1707, 
under the title of ARITHMETICS UNIVERSALIS, SINE DE COMPOSI 
TIONS ET RESOLUTIONS ARITHMETICA LIBER. This publication 
is said to have been a breach of confidence on Whiston s part. Mr. 
Ralphson, not long afterward, translated the work into English ; 
and a second edition of it, with improvements by the author, was 
issued at London, 1712, by Dr. Machin. Subsequent editions, 
both in English and Latin, with commentaries, have been published. 

In June, 1709, Newton intrusted the superintendence of a 
second edition of the PRINCIPIA to Roger Cotes, Plumian Pro 
fessor of Astronomy at Cambridge. The first edition had been 
sold off for some time. Copies of the work had become very 
rare, and could only be obtained at several times their original 
cost. A great number of letters passed oetween the author and 
Mr. Cotes during the preparation of the edition, which finally 
appeared in May, 1713. It had many alterations and improve 
ments, and was accompanied by an admirable Preface from the 
pen of Cotes. 

Our author s early Treatise, entitled, ANALYSIS PER EQUATIONES 
NUMERO TERMINORUM INFINITAS, as well as a small Tract, Gearing 
the title of METHODUS DIFFERENTIALS, was published, witn nis 
consent, in 1711. The former of these, and the Treatise De 
Quadratura Curvarum, translated into Englisn, witn a .arge com 
mentary, appeared in 1745. His work, entitled. ARTIS ANA 
LYTICS SPECIMINA, VEL GEOMETRIA ANALYTICA, was iirs; given 
to the world in the edition of Dr. Horsley, 1779. 



48 LIFE OF SIR ISAAC NEWTON. 

It is a notable fact, in Newton s history, that he never volun* 
tarily published any one of his purely mathematical writings 
The cause of this unwillingness in some, and, in other instances, 
of his indifference, or, at least, want of solicitude to put forth his 
works may be confidently sought for in his repugnance to every 
thing like contest or dispute. But, going deeper than this aver 
sion, we find, underlying his whole character and running parallel 
with all his discoveries, that extraordinary humility which always 
preserved him in a position so relatively just to the behests of 
time and eternity, that the infinite value of truth, and the utter 
worthlessness of fame, were alike constantly present to him. 
Judging of his course, however, in its more temporary aspect, as 
bearing upon his immediate quiet, it seemed the most unfortunate. 
For an early publication, especially in the case of his Method of 
Fluxions, would have anticipated all rivalry, and secured him 
from the contentious claims of Leibnitz. Still each one will solve 
the problem of his existence in his own way, and, with a manlike 
Newton, his own, as we conceive, could be no other than the best 
way. The conduct of Leibnitz in this affair is quite irreconcilable 
with the stature and strength of the man ; giant-like, and doing 
nobly, in many ways, a giant s work, yet cringing himself into the 
dimensions and performances of a common calumniator. Opening 
in 1699, the discussion in question continued till the close of 
Leibnitz s life, in 1716. We give the summary of the case as 
contained in the Report of the Committee of the Royal Society, 
the deliberately weighed opinion of which has been adopted as an 
authoritative decision in all countries. 

" We have consulted the letters and letter books in the custody 
of the Royal Society, and those found among the papers of Mr. 
John Collins, dated between the years 1669 and 1677, inclusive ; 
and showed them to such as knew and avouched the hands of Mr. 
Barrow, Mr. Collins, Mr. Oldenburg, and Mr. Leibnitz ; and 
compared those of Mr. Gregory with one another, and with copies 
of some of them taken in the hand of Mr. Collins ; and have 
extracted from them what relates to the matter referred to us : 
all which extracts, herewith delivered to you, we believe to be 
genuine and authentic. And by these letters and papers wf 
find: 



LIFE OF SIR ISAAC NEWTON. 49 

" I. Mr. Leibnitz was in London in the beginning of the year 
1673 ; and went thence in or about March, to Paris, where he 
kept a correspondence with Mr. Collins, by means of Mr. Olden 
burg, till about September, 1676, and then returned, by London 
and Amsterdam, to Hanover: and that Mr. Collins was very free 
in communicating to able mathematicians what he had received 
from Mr, Newton and Mr. Gregory. 

" II. That when Mr. Leibnitz was the first time in London, 
he contended for the invention of another differential method, 
properly so called ; and, notwithstanding he was shown by Dr. 
Pell that it was Newton ? s method, persisted in maintaining it to 
be his own invention, by reason that he had found it by himself 
without knowing what Newton had done before, and had much 
improved it. And we find no mention of his having any other 
differential method than Newton s before his letter of the 21st of 
June, 1677, which was a year after a copy of Mr. Newton s letter 
of the 10th of December, 1672, had been sent to Paris to be 
communicated to him ; and above four years after Mr. Collins 
began to communicate that letter to his correspondents ; in which 
letter the method of fluxions was sufficiently described to any 
intelligent person. 

"III. That by Mr. Newton s letter, of the 13th of June, 1676 
it appears that he had the method of fluxions above five years 
before the writing of that letter. And by his Analysis per ^Equa- 
tiones numero Terminorum Infmitas, communicated by Dr. Barrow 
to Mr. Collins, in July, 1669, we find that he had invented the 
method before that time. 

"IV. That the differential method is one and the same with 
the method of fluxions, excepting the name and mode of notation ; 
Mr. Leibnitz calling those quantities differences which Mr. Newton 
calls moments, or fluxions ; and marking them with a letter d a 
mark not used by Mr. Newton. 

" And, therefore, we take the proper question to be, not who 
invented this or that method, but, who was the first inventor of 
the method ? And we believe that those who have reputed Mr. 
Leibnitz the first inventor knew little or nothing of his correspond 
ence with Mr. Collins and Mr. Oldenburg long before, nor of Mr. 



50 LIFE OP SIR ISAAC NEWTON. 

Newton s hiving that method above fifteen years before Mr 
Leibnitz began to publish it in the Acta Eruditorum of Leipsic. 

" For which reason we reckon Mr. Newton the first inventor ; 
and are of opinion that Mr. Keill, in asserting the same, has been 
no ways injurious to Mr. Leibnitz. And we submit to the judg 
ment of the Society, whether the extract and papers, now pre 
sented to you, together with what is extant, to the same pur 
pose, in Dr. Wallis s third volume, may not deserve to be made 
public." 

This Report, with the collection of letters and manuscripts, 
under the title of COMMERCIUM EPISTOLICUM D. JOHANNIS COLLINS 

ET ALIORUM DE ANALYSI PROMOTA JuSSU SoCIETATIS REGIES 

EDITUM, appeared accordingly in the early part of 1713. Its 
publication seemed to infuse additional bitterness into the feelings 
of Leibnitz, who descended to unfounded charges and empty 
threats. He had been privy counsellor to the Elector of Han 
over, before that prince was elevated to the British throne ; and 
in his correspondence, in 1715 and 1716, with the Abbe Conti, 
then at the court of George L, and with Caroline, Princess of 
Wales, he attacked the doctrines of the PRINCIPIA, and indirectly 
its author, in a manner very discreditable to himself, both as a 
learned and as an honourable man. His assaults, however, were 
triumphantly met; and, to the complete overthrow of his rival 
pretensions, Newton was induced to give the finishing blow. The 
verdict is universal and irreversible that the English preceded 
the German philosopher, by at least ten years, in the invention 
of fluxions. Newton could not have borrowed from Leibnitz ; 
but Leibnitz might have borrowed from Newton. A new edition 
of the Commercium Epistolicum was published in 1722-5 (?) ; but 
neither in this, nor in the former edition, did our author take any 
part. The disciples, enthusiastic, capable and ready, effectually 
shielded, with the buckler of Truth, the character of the Master, 
whose own conduct throughout was replete with delicacy, dignity 
and justice. He kept aloof from the controversy in which Dr. 
Keill stood forth as the chief representative of the Newtonian 
side till the very last, when, for the satisfaction of the King, 
George L. rather than for his own, he consented to put forth his 



LIFE OF SI| L^.-vJ NEWTON. 5i 

hand and firmly secure his rights upon a certain and impregnable, 
basis. 

A petition to have inventions for promoting the discovery of the 
longitude at sea, suitably rewarded, was presented to the House 
of Commons, in 1714. A committee, having been appointed to 
investigate the subject, called upon Newton and others for their 
opinions. That of our author was given in writing, A report, 
favourable to the desired measure, was then taken up, and a bill 
for its adoption subsequently passed. 

On the ascension of George I., in 1714, Newton became an 
object of profound interest at court. His position under govern 
ment, his surpassing fame, his spotless character, and. above all, 
his deep and consistent piety, attracted the reverent regard of the 
Princess of Wales, afterward queen -consort to George II. She 
was a woman of a highly cultivated mind, and derived the greatest 
pleasure from conversing with Newton and corresponding with 
Leibnitz. One day, in conversation with her, our author men 
tioned and explained a new system of chronology, which he had 
composed at Cambridge, where he had been in the habit " of 
refreshing himself with history and chronology, when he wa c 
weary with other studies." Subsequently, in the year 1718, she 
requested a copy of this interesting and ingenious work Newton, 
accordingly, drew up an abstract of the system from the separate 
papers in which it existed, and gave it to her on condition that it 
should riot be communicated to any other person. Sometime 
afterward she requested that the Abbe Conti might be allowed 
to have a copy of it The author consented: and the abbe 
received a copy of the manuscript, under the like injunction and 
promise of secrecy. This manuscript bore the title of " A short 
Chronicle, from the First Memory of Tilings in Europe, to the 
Conquest of Persia, by Alexander the Great." 

After Newton took up his residence in London, he lived in a 
style suited to his elevated position and rank. He kept his car 
riage, with an establishment of three male and three female serv 
ants. But to everything like vain show and luxury he was utterly 
averse. His household affairs, for the last twenty years of his 
life, were under the charge of his niece, Mrs. Catherine Barton, 



52 LIFE OF SIR ISAAC NEWTON. 

wife and widow of Colonel Barton a woman of great beauty and 
accomplishment and subsequently married to John Conduit, Esq. 
At home Newton was distinguished by that dignified and gentle 
hospitality which springs alone from true nobleness. On all pro 
per occasions, he gave splendid entertainments, though without 
ostentation. In society, whether of the palace or the cottage, 
his manner was self-possessed and urbane ; his look benign and 
affable ; his speech candid and modest ; his whole air undisturb 
edly serene. He had none of what are usually called the singu 
larities of genius ; suiting himself easily to every company 
except that of the vicious and wicked ; and speaking of himself 
and others, naturally, so as never even to be suspected of vanity. 
There was in him, if we may be allowed the expression, a WHOLE 
NESS of nature, which did not admit of such imperfections and 
weakness the circle was too perfect, the law too constant, and 
the disturbing forces too slight to suffer scarcely any of those 
eccentricities which so interrupt and mar the movements of many 
bright spirits, rendering their course through the world more like 
that of the blazing meteor than that of the light and life-impart 
ing sun. In brief, the words GREATNESS and GOODNESS could 
not, humanly speaking, be more fitly employed than when applied 
as the pre-eminent characteristics of this pure, meek and vene 
rable sage. 

In the eightieth year of his age, Newton was seized with 
symptoms of stone in the bladder. His disease was pronounced 
incurable. He succeeded, however, by means of a strict regimen, 
and other precautions, in alleviating his complaint, and procuring 
long intervals of ease. His diet, always frugai, was now extremely 
temperate, consisting chiefly of broth, vegetables, and fruit, with, 
now and then, a little butcher meat. He gave up the use of his 
carriage, and employed, in its stead, when he went out, a chair. 
All invitations to dinner were declined ; and only small parties 
were received, occasionally, at his own house. 

In 1724 he wrote to the Lord Provost of Edinburgh, offering 
to contribute twenty pounds yearly toward the salary of Mr. 
Maclaurin, provided he accepted the assistant Professorship of 
Mathematics in the University of that place. Not only in the 



LIFE OP SIR ISAAC NEWTON. 53 

cause of ingenuity and learning, but in that of religion in relieving 
the poor and .assisting his relations, Newton annually expended 
large sums. He was generous and charitable almost to a fault. 
Those, he would often remark, who gave away nothing till they 
died, never gave at all. His wealth had become considerable by 
a prudent economy ; but he regarded money in no other light 
than as one of the means wherewith he had been intrusted to do 
good, and he faithfully employed it accordingly. 

He experienced, in spite of all his precautionary measures, a 
return of his complaint in the month of August, of the same year, 
1 724, when he passed a stone the size of pea ; it came from him 
in two pieces, the one at the distance of two day.s from the other. 
Tolerable good health then followed for some months. In Janu 
ary, 1725, however, he was taken with a violent cough and inflam 
mation of the lungs. In consequence of this attack, he was pre 
vailed upon to remove to Kensington, where his health greatly 
improved. In February following, he was attacked in both feet 
with the gout, of the approach of which he had received, a few 
years before, a slight warning, and the presence of which now 
produced a very beneficial change in his general health. Mr. 
Conduit, his nephew, has recorded a curious conversation which 
took place, at or near this time, between himself and Sir Isaac. 

"I was, on Sunday night, the 7th March, 1724-5, at Kensing 
ton, with Sir Isaac Newton, in his lodgings, just after he was out 
of a fit of the gout, which he had had in both of his feet, for the 
first time, in the eighty-third year of his age. He was better after 
it, and his head clearer and memory stronger than I had known 
them for some time. He then repeated to me, by way of dis 
course, very distinctly, though rather in answer to my queries, 
than in one continued narration, what he had often hinted to me 
before, viz. : that it was his conjecture (he would affirm nothing) 
that there was a sort of revolution in the heavenly bodies ; that 
the vapours and light, emitted by the sun, which had their sedi 
ment, as water and other matter, had gathered themselves, by 
degrees, into a body, and attracted more matter from the planets, 
and at last made a secondary planet (viz. : one of those that go 
round another planet), and then, by gathering to them, and 



54 LIFE OF SIR ISAAC NEWTON. 

attracting more matter, became a primary planet ; and then, bf 
increasing still, became a comet, which, after certain revolutions, 
by coming nearer and nearer to the sun, had all its volatile parts 
condensed, and became a matter tit to recruit and replenish the 
sun (which must waste by the constant heat and light it emitted), 
as a faggot would this fire if put into it (we were sitting by a 
wood fire), and that that would probably be the effect of the 
comet of 1680, sooner or later ; for, by the observations made 
upon it, it appeared, before it came near the sun, with a tail only 
two or three degrees long ; but, by the heat it contracted, in going 
so near the sun, it seemed to have a tail of thirty or forty degrees 
when it went frpm it ; that he could not say when this comet 
would drop into the sun ; it might perhaps have five or six revo 
lutions more first, but whenever it did it would so much increase 
the heat of the sun that this earth would be burned, and no ani 
mals in it could live. That he took the three phenomena, seen 
by Hipparchus, Tycho Brahe, and Kepler s disciples, to have been 
of this kind, for he could not otherwise account for an extraor 
dinary light, as those were, appearing, all at once, among the 
the fixed stars (all which he took to be suns, enlightening other 
planets, as our sun does ours), as big as Mercury or Venus seems 
to us, and gradually diminishing, for sixteen months, and then 
sinking into nothing. He seemed to doubt whether there were 
not intelligent beings, superior to us, who superintended these 
revolutions of the heavenly bodies, by the direction of the Supreme 
Being. He appeared also to be very clearly of opinion that the 
inhabitants of this world were of short date, and alledged, as one 
reason for that opinion, that all arts, as letters, ships, printing, 
needle, &c., were discovered within the memory of history, which 
could not have happened if the world had been eternal ; and that 
there were visible marks of ruin upon it which could not be 
effected by flood only. When I asked him how this earth could 
have been repeopled if ever it had undergone the same fate 
it was threatened with hereafter, by the comet of 1680, he 
answered, that required the power of a Creator. He said he 
took all the planets to be composed of the same matter with this 
earth, viz. : earth, water, stones, &c. 3 but variously concocted. J 



LIFE OP SIR ISAAC NEWTON. 55 

asked him why he would not publish his conjectures, as conjec 
tures, and instanced that Kepler had communicated his ; and 
though he had not gone near so far as Kepler, yet Kepler s 
guesses were so just and happy that they had been proved and 
demonstrated by him. His answer was, " I do not deal in con 
jectures." But, on my talking to him about the four observations 
that had been made of the comet of 1680, at 574 years distance, 
and asking him the particular times, he opened his Principia, 
which laid on the table, and showed me the particular periods, 
viz.: 1st. The Julium Sidus, in the time of Justinian, in 1106, 
in 1680. 

" And I, observing that he said there of that comet, incidet 
in corpus solis, and in the next paragraph adds, stellae fixae 
refici possunt, told him I thought he owned there what we had 
been talking about, viz. : that the comet would drop into the sun, 
and that fixed stars were recruited and replenished by comets 
when they dropped into them ; and, consequently, that the sun 
would be recruited too ; and asked him why he would not own as 
fully what he thought of the sun as well as what he thought of 
the fixed stars. He said, that concerned us more; and, laugh 
ing, added, that he had said enough for people to know his 
meaning." 

In the summer of 1725, a French translation of the chronolo 
gical MS., of which the Abbe Conti had been permitted, some 
time previous, to have a copy, was published at Paris, in violation 
of all good faith. The Punic Abbe had continued true to his 
promise of secrecy while he remained in England ; but no sooner 
did he reach Paris than he placed the manuscript into the hands 
of M. Freret, a learned antiquarian, who translated the work, and 
accompanied it with an attempted refutation of the leading points 
of the system. In November, of the same year, Newton received 
a presentation copy of this publication, which bore the title of 
ABREGE DE CHRONOLOGIE DE M. LE CHEVALIER NEWTON, FAIT 

PAR LUI-MEME, ET TRADUIT SUR LE MANUSCRIPT ANGLAIS. Soon 

afterward a paper entitled, REMARKS ON TFE OBERVATIONS MADE 
ON A CHRONOLOGICAL INDEX OF SIR ISAAC NE.WTON, TRANSLATED 
INTO FRENCH BY THE OBSERVATOR, ANL PUBLISHED AT PARIS, 



56 LIFE OF SIR ISAAC NEWTON, 

was drawn up by our author, and printed in the Philosophical 
Transactions for 1725. It contained a history of the whole 
matter, and a triumphant reply to the objections of M. Freret. 
This answer called into the field a fresh antagonist, Father Soueiet, 
whose five dissertations on this subject were chiefly remarkable 
for the want of knowledge and want of decorum, which they 
displayed. In consequence of these discussions, Newton was in 
duced to prepare his larger work for the press, and had nearly 
completed it at the time of his death. It was published in 1728, 
under the title of THE CHRONOLOGY OF THE ANCIENT KINGDOMS 
AMENDED, TO WHICH is PREFIXED A SHORT CHRONICLE FROM THE 

FIRST MEMORY OF THINGS IN EUROPE TO THE CONQUEST OF 

PERSIA BY ALEXANDER THE GREAT. It consists of six chap 
ters: 1. On the Chronology of the Greeks; according to Whis- 
ton, our author wrote out eighteen copies of this chapter with his 
own hand, differing little from one another. 2. Of the Empire 
of Egypt; 3. Of the Assyrian Empire; 4. Of the two contempo 
rary Empires of the Babylonians and Medes ; 5. A Description 
of the Temple of Solomon ; 6. Of the Empire of the Persians ; 
this chapter was not found copied with the other five, but as it 
was discovered among his papers, arid appeared to be a continu 
ation of the same work, the Editor thought proper to add it 
thereto. Newton s LETTER TO A PERSON OF DISTINCTION WHO 

HAD DESIRED HIS OPINION OF THE LEARNED BlSHO^ LLOYD S 

HYPOTHESIS CONCERNING THE FORM OF THE MOST ANCIENT 
^EAR, closes this enumeration of his Chronological Writings. 

A ihird edition of the PRINCIPIA appeared in 1726, with many 
changes and additions. About four years were consumed in its 
preparation and publication, which were under the superintend- 
ance of Dr. Henry Pemberton, an accomplished mathematician, 
and the author of "A VIEW OF SIR ISAAC NEWTON S PHILO 
SOPHY." 1728. This gentleman enjoyed numerous opportunities 
of conversing with the aged and illustrious author. " I found," 
says Pemberton, " he had read fewer of the modern mathemati 
cians than one could have expected; but his own prodigious 
invention readily supplied him with what he might have an occa 
sion for in the pursuit of any subject he undertook. I have often 



LIFE OF SIR ISAAC NEWTON. 57 

heard him censure the handling geometrical subjects ly algebraic 
calculations ; and his book of Algebra he called by the name of 
Universal Arithmetic, in opposition to the injudicious title of 
Geometry, which Descartes had given to the treatise, wherein he 
shows how the geometer may assist his invention by such kind 
of computations. He thought Huygens the most elegant of any 
mathematical writer of modern times, and the most just imitator 
of the ancients. Of their taste and form of demonstration, Sir 
Isaac always professed himself a great admirer. I have heard 
him even censure himself for not following them yet more closely 
than he did ; and speak with regret of his mistake at the begin 
ning of his mathematical studies, in applying himself to the works 
of Descartes and other algebraic writers, before he had considered 
the elements of Euclid with that attention which so excellent a 
writer deserves." 

" Though his memory was much decayed," continues Dr. Pem- 
berton, "he perfectly understood his own writings." And even 
this failure of memory, we would suggest, might have been more 
apparent than real, or, in medical terms, more the result of func 
tional weakness than organic decay. Newton seems never to 
have confided largely to his memory : and as this faculty mani 
fests the most susceptibility to cultivation ; so, in the neglect of 
due exercise, it more readily and plainly shows a diminution of 
its powers. 

Equanimity and temperance had, indeed, preserved Newton 
singularly free from all mental and bodily ailment. His hair was, 
to the last, quite thick, though as white as silver. He never 
made use of spectacles, and lost but one tooth to the day of his 
death. He was of middle stature, well-knit, and, in the latter 
part of his life, somewhat inclined to be corpulent. Mr. Conduit 
says, " he had a very lively and piercing eye, a comely and gra 
cious aspect, with a fine head of hair, white as silver, without any 
baldness, and when his peruke was off was a venerable sight." 
According to Bishop Atterbury, "in the whole air of his face and 
make there was nothing of that penetrating sagacity which 
appears in his compositions. He had something rather languid 
in his look and manner which did not raise any great expectation 



58 LIFE OF SIR ISAAC NEWTON. 

in those who did not know him." Hearne remarks, " Sir Isaac 
was a man of no very promising aspect. He was a short, well- 
set man. He was full of thought, and spoke very little in com 
pany, so that his conversation was not agreeable. When he rode 
in his coach, one arm would be out of his coach on one side and 
the other on the other." These different accounts we deem 
easily reconcilable. In the rooms of the Royal Society, in the 
street, or in mixed assemblages, Newton s demeanour always 
courteous, unassuming and kindly still had in it the overawings 
of a profound repose and reticency, out of which the communica 
tive spirit, and the "lively and piercing eye" would only gleam 
in the quiet and unrestrained freedom of his own fire-side. 

" But this I immediately discovered in him," adds Pemberton, 
still further, "which at once both surprised and charmed me. 
Neither his extreme great age, nor his universal reputation had 
rendered him stiff in opinion, or in any degree elated. Of this I 
had occasion to have almost daily experience. The remarks I 
continually sent him by letters on his Principia, were received 
with the utmost goodness. These were so far from being any 
ways displeasing to him, that, on the contrary, it occasioned him 
to speak many kind things of me to my friends, and to honour me 
with a public testimony of his good opinion." A modesty, open 
ness, and generosity, peculiar to the noble and comprehensive 
spirit of Newton. " Full of wisdom and perfect in beauty," yet 
not lifted up by pride nor corrupted by ambition. None, how 
ever, knew so well as himself the stupendousness of his discoveries 
in comparison with all that had been previously achieved ; and 
none realized so thoroughly as himself the littleness thereof in 
comparison with the vast region still unexplored. A short time 
before his death he uttered this memorable sentiment: " I do not 
know what I may appear to the world ; but to myself I seem to 
have been only like a boy playing on the sea-shore, and diverting 
myself in now and then finding a smoother pebble or a prettier 
shell than ordinary, while the great ocean of truth lay all undis 
covered before me." How few ever reach the shore even, much 
less find "a smoother pebble or a prettier shell!" 

Newton had now resided about two years at Kensington ; and 



LIFE OF SIR ISAAC NEWTON. 59 

the air which he enjoyed there, and the state of absolute rest, 
proved of great benefit to him. Nevertheless he would occasion 
ally go to town. And on Tuesday, the 28th of February, 1727, 
he proceeded to London, for the purpose of presiding at a meeting 
of the Royal Society. At this time his health was considered, 
by Mr. Conduit, better than it had been for many years. But 
the unusual fatigue he was obliged to suffer, in attending the 
meeting, and in paying and receiving visits, speedily produced a 
violent return of the affection in the bladder. He returned to 
Kensington on Saturday, the 4th of March. Dr. Mead and Dr. 
Cheselden attended him ; they pronounced his disease to be the 
stone, and held out no hopes of recovery. On Wednesday, the 
15th of March, he seemed a little better; and slight, though 
groundless, encouragement was felt that he might survive the 
attack. From the very first of it, his sufferings had been intense. 
Paroxysm followed paroxysm, in quick succession : large drops 
)f sweat rolled down his face ; but not a groan, not a complaint, 
not the least mark of peevishness or impatience escaped him : 
and during the short intervals of relief, he even smiled and con 
versed with his usual composure and cheerfulness. The flesh 
quivered, but the heart quaked not ; the impenetrable gloom was 
settling down : the Destroyer near ; the portals of the tomb 
opening, still, arnid this utter wreck and dissolution of the mortal, 
the immortal remained serene, unconquerable : the radiant light 
broke through the gathering darkness ; and Death yielded up its 
sting, and the grave its victory. On Saturday morning, 18th, 
he read the newspapers, and carried on a pretty long conversation 
with Dr. Mead. His senses and faculties were then strong and 
vigorous ; but at six o clock, the same evening, he became insen 
sible ; and in this state he continued during the whole of Sunday, 
and till Monday, the 20th, when he expired, between one and 
two o clock in the morning, in the eighty-fifth year of his age. 

And these were the last days of Isaac Newton. Thus closed 
the career of one of earth s greatest and best men. His mission 
was fulfilled. Unto the Giver, in many-fold addition, the talents 
were returned. While it was yet day he had worked ; and for 
the night that quickly cometh he was not unprepared. Full of 



60 LIFE OF SIR ISAAC NEWTON. 

years, ind full of honours, the heaven-sent was recalled ; and, in 
the confidence of a " certain hope," peacefully he passed awa} 
into the silent depths of Eternity. 

His body was placed in Westminster Abbey, with the state 
and ceremonial that usually attended the interment of the most 
distinguished. In 1731, his relatives, the inheritors of his personal 
estate, erected a monument to his memory in the most conspicu 
ous part of the Abbey, which had often been refused by the dean 
and chapter to the greatest of England s nobility. During the 
same year a medal was struck at the Tower in his honour ; arid, 
in 1755, a full-length statue of him, in white marble, admirably 
executed, by Roubiliac, at the expense of Dr. Robert Smith, was 
erected in the ante-chamber of Trinity College, Cambridge. 
There is a painting executed in the glass of one of the windows 
of the same college, made pursuant to the will of Dr. Smith, who 
left five hundred pounds for that purpose. 

Newton left a personal estate of about thirty-two thousand 
pounds. It was divided among his four nephews and four nieces 
of the half blood, the grand-children of his mother, by the Reve 
rend Mr. Smith. The family estates of Woolsthorpe arid Sustern 
fell to John Newton, the heir-at-law, whose great grand-father 
was Sir Isaac s uncle. Before his death he made an equitable 
distribution of his two other estates : the one in Berkshire to the 
sons and daughter of a brother of Mrs. Conduit ; and the other, 
at Kensington, to Catharine, the only daughter of Mr. Conduit, 
and who afterward became Viscountess Lymington. Mr. Con 
duit succeeded to the offices of the Mint, the duties of which he 
had discharged during the last two years of Sir Isaac s life. 

Our author s works are found in the collection of Castilion, 
Berlin, 1744, 4to. 8 torn.; in Bishop Horsley s Edition, London, 
1779, 4to. 5 vol.; in the Biographia Brittannica, &c. Newton 
also published Bern. Varcnii Geographia, &c., 1681, 8vo. 
There are, however, numerous manuscripts, letters, and other 
papers, which have never been given to the world: these are 
preserved, in various collections, namely, in the library of Trinity 
College, Cambridge ; in the library of Corpus Christi College, 
Oxford ; in the library of Lord Macclesfield : and, lastly arid 



LIFE OF SIR ISAAC NEWTON. 61 

chiefly, in the possession of the family of the Earl of Portsmouth, 
through the Viscountess Lymington. 

Everything appertaining to Newton has been kept and che 
rished with peculiar veneration. Different memorials of him are 
preserved in Trinity College, Cambridge ; in the rooms of the 
Royal Society, of London : and in the Museum of the Royal 
Society of Edinburgh. 

The manor-house, at Woolsthorpe, was visited by Dr. Stuke 
ley, in October, 1721, who, in a letter to Dr. Mead, written in 
1727, gave the following description of it: " Tis built of stone, 
as is the way of the country hereabouts, and a reasonably good 
one. They led me up stairs and showed me Sir Isaac s stud}-, 
where I supposed he studied, when in the country, in his younger 
days, or perhaps when he visited his mother from the University. 
I observed the shelves were of his own making, being pieces of 
deal boxes, which probably he sent his books and clothes down 
in on those occasions. There were, some years ago, two or threr 
hundred books in it of his father-in-law, Mr. Smith, which Sir 
Isaac gave to Dr. Newton, of our town." The celebrated apple- 
tree, the fall of one of the apples of which is said to have turned 
the attention of Newton to the subject of gravity, was destroyed 
by the wind about twenty years ago ; but it has been preserved 
in the form of a chair. The house itself has been protected with 
religious care. It was repaired in 1798, and a tablet of white 
marble put up in the room where our author was born, with the 
follow, ng inscription : 

" Sir Isaac Newton, son of John Newton, Lord of the Manor 
of Woolsthorpe, was born in this room, on the 25th of December, 
1642." 

Nature and Nature s Laws wei-e hid in night, 
God said, " Let NEWTON be," and all was light. 



THE PEINCIPIA. 



THE AUTHOR S PREFACE 

SINCE the ancients (as we are told by Pappus), made great account oi 
the science of mechanics in the investigation of natural things : and the 
moderns, laying aside substantial forms and occult qualities, have endeav 
oured to subject the phenomena of nature to the laws of mathematics, I 
have in this treatise cultivated mathematics so far as it regards philosophy. 
The ancients considered mechanics in a twofold respect ; as rational, which 
proceeds accurately by demonstration ; and practical. To practical me 
chanics all the manual arts belong, from which mechanics took its name. 
Rut as artificers do not work with perfect accuracy, it comes to pass that 
mechanics is so distinguished from geometry, that what is perfectly accu 
rate is called geometrical , what is less so, is called mechanical. But the 
errors are not in the art, but in the artificers. He that works with less 
accuracy is an imperfect mechanic ; and if any could work with perfect 
accuracy, he would be the most perfect mechanic of all ; for the description 
if right lines and circles, upon which geometry is founded, belongs to me 
chanics. Geometry does not teach us to draw these lines, but requires 
them to be drawn ; for it requires that the learner should f.rst be taught 
to describe these accurately, before he enters upon geometry ; then it shows 
how by these operations problems may be solved. To describe right lines 
and circles are problems, but not geometrical problems. The solution of 
these problems is required from mechanics ; and by geometry the use of 
them, when so solved, is shown ; and it is the glory of geometry that from 
those few principles, brought from without, it is able to produce so many 
things. Therefore geometry is founded in mechanical practice, and is 
nothing but that part of universal mechanics which accurately proposes 
and demonstrates the art of measuring. But since the manual arts are 
chiefly conversant in the moving of bodies, it comes to pass that geometry 
is commonly referred to their magnitudes, and mechanics to their motion. 
In this sense rational mechanics will be the science of motions resulting 
from any forces whatsoever, and of the forces required to produce any mo 
tions, accurately proposed and demonstrated. This part of mechanics was 



i:;vm THE AUTHOR & PREFACE. 

cultivated by the ancients in the five powers which relate to manual arts, 
who considered gravity (it not being a manual power), ho Otherwise than 
as it moved weights by those powers. Our design not respecting arts, but 
philosophy, and our subject not manual but natural powers, we consider 
chiefly those things which relate to gravity, levity, elastic force, the resist 
ance of fluids, and the like forces, whether attractive or impulsive ; and 
therefore we offer this work as the mathematical principles :f philosophy ; for 
all the difficulty of philosophy seems to consist in this from the phenom 
ena of motions to investigate the forces of nature, and then from these 
forces to demonstrate the other phenomena ; and to this end the general 
propositions in the first and second book are directed. In the third book 
we give an example of this in the explication of the System of the World : 
for by the propositions mathematically demonstrated in the former books, 
we in the third derive from the celestial phenomena the forces of gravity 
with which bodies tend to the sun and the several planets. Then from these 
forces, by other propositions which are also mathematical, we deduce the mo 
tions of the planets, the comets, the moon, and the sea. I wish we could do- 
rive the rest of the phenomena of nature by the same kind of reasoning from 
mechanical principles; for I am induced by many reasons to suspect that 
they may all depend upon certain forces by which the particles of bodies. 
by some causes hitherto unknown, are either mutually impelled towards 
each other, and cohere in regular figures, or are repelled and recede from 
each other; which forces being unknown, philosophers have hitherto at 
tempted the search of nature in vain ; but I hope the principles here laid 
down will afford some light either to this or some truer method of philosophy. 
In the publication of this work the most acute and universally learned 
Mr. Edmund H alley not only assisted me with his pains in correcting the 
press and taking care of the schemes, but it was to his solicitations that its 
becoming public is owing ; for when he had obtained of me my demonstra 
tions of the figure of the celestial orbits, he continually pressed me to com 
municate the same to the Royal Societ //, who afterwards, by their kind en 
couragement and entreaties, engaged me to think of publishing them. But 
after I had begun to consider the inequalities of the lunar motions, and 
had entered upon some other things relating to the laws and measures oi 
gravity, and other forces : and the figures that would be described by bodies 
attracted according to given laws ; and the motion of several bodies moving 
among themselves; the motion of bodies in resisting mediums; the forces, 
densities, and motions, of rn( Hums ; the orbits of the comets, and such like ; 



Ixix 

deferred that publication till I had made a searcli into those matters, and 
could put forth the whole together. What relates to the lunar motions (be 
ing imperfect), I have put all together in the corollaries of Prop. 66, to 
avoid being obliged to propose and distinctly demonstrate the several things 
there contained in a method more prolix than the subject deserved, and in 
terrupt the series of the several propositions. Some things, found out after 
the rest, I chose to insert in places less suitable, rather than change the 
number of the propositions and the citations. I heartily beg that what 1 
have here done may be read with candour; and that the defects in a 
subject so difficult be not so much reprehended as kindly supplied, and in 
vestigated by new endeavours of mv readers. 

ISAAC NEWTON. 

Cambridge, Trinity Coupge May 8, liHB. 

In the second edition the second section of the first book was enlarged. 
In the seventh section of the second book the theory of the resistances of fluids 
was more accurately investigated, and confirmed by new experiments. In 
the third book the moon s theory and the profession of the equinoxes were 
more fully deduced from their principles ; and the theory of the comets 
was confirmed by more examples of the calculati >n of their orbits, done 
also with greater accuracy. 

In this third edition the resistance of mediums is somewhat more largely 
handled than before; and new experiments of the resistance of heavy 
bodies falling in air are added. In the third book, the argument to prove 
that the moon is retained in its orbit by the force of gravity is enlarged 
on ; and there are added new observations of Mr. Pound s of the proportion 
of the diameters of Ju.piter to each other : there are, besides, added Mr. 
Kirk s observations of the comet in 16SO ; the orbit of that comet com 
puted in an ellipsis by Dr. Halley ; and the ortit of the comet in 
computed by Mr. Bradley, 



OOK I. 



THE 

MATHEMATICAL PRINCIPLES 



OF 



NATURAL PHILOSOPHY 



DEFINITIONS. 

DEFINITION I. 
77w? quantity of matter is the measure of the same, arising from its 

density and hulk conjutictly. 

THUS air of a double density, in a double space, is quadruple in quan- 
ti ty ; in a triple space, sextuple in quantity. The same thing is to be un 
derstood of snow, and fine dust or powders, that are condensed by compres 
sion or liquefaction and of all bodies that are by any causes whatever 
differently condensed. I have no regard in this place to a medium, if any 
such there is, that freely pervades the interstices between the parts oi 
bodies. It is this quantity that I mean hereafter everywhere under the 
name of body or mass. And the same is known by the weight of each 
body ; for it is proportional to the weight, as I have found by experiments 
on pendulums, very accurately made, which shall be shewn hereafter. 

DEFINITION II. 

The quantity of motion is the measure nf tlie same, arising from the 

velocity and quantity of matter corjunctly. 

The motion of the whole i<! the sum of the motions of all the parts ; and 
therefore in a body double in quantity, with equal velocity, the motion is 
iouble ; with twice the velocity, it is quadruple, 

DEFINITION III. 

The vis insita, or innate force of matter, is a power of resisting, hy 
which every body, as much as in it lies, endeavours to persevere in its 
present stale, whether it be of rest, or of moving uniformly forward 
in a right line. 
This force is ever proportional to the body whose force it is ; and differs 

nothing from the inactivity of the mass, but in our manner of conceiving 



T4 THE MATHEMATICAL PRINCIPLES 

it. A body, from the inactivity of matter, is not without difficulty put out 
of its state of rest or motion. Upon which account, this vis insita, may, 
by a most significant name, be called vis inertia, or force of inactivity. 
Hut a body exerts this force only, when another force, impressed upon it, 
endeavours to change its condition ; and the exercise of this force may bo 
considered both as resistance and impulse ; it is resistance, in so far as the 
body, for maintaining its present state, withstands the force impressed; it 
is impulse, in so far as the body, by not easily giving way to the impressed 
force of another, endeavours to change the state of that other. Resistance 
is usually ascribed to bodies at rest, and impulse to those in motion; 
but motion and rest, as commonly conceived, are only relatively distin 
guished ; nor are those bodies always truly at rest, which commonly are 
taken to be so. 

DKFLMTIOX IV. 

Ait impressed force is an action exerted upon a body, in order to change 
its state, either of rest, or of moving uniformly forward in a right 
line. 

This force consists in the action only; and remains no longer in the 
body, when the action is over. For a body maintains every new state it 
acquires, by its vis inertice only. Impressed forces are of differe.it origins 
as from percussion, from pressure, from centripetal force. 

DEFINITION V. 

A centripetal force is that by which bodies are drawn or impelled, or any 

way tend, towards a point as to a centre. 

Of this sort is gravity, by which bodies tend to the centre of the earth 
magnetism, by which iron tends to the loadstone ; and that force, what 
ever it is, by which the planets are perpetually drawn aside from the rec 
tilinear motions, which otherwise they would pursue, and made to revolve 
in curvilinear orbits. A stone, whirled about in a sling, endeavours to re 
cede from the hand that turns it ; and by that endeavour, distends the 
sling, and that with so much the greater force, as it is revolved with the 
greater velocity, and as soon as ever it is let go, flies away. That force 
which opposes itself to this endeavour, and by which the sling perpetually 
draws back the stone towards the hand, and retains it in its orbit, because 
it is directed to the hand as the centre of the orbit, I call the centripetal 
force. And the same thing is to be understood of all bodies, revolved in 
any orbits. They all endeavour to recede from the centres of their orbits ; 
and wore it not for the opposition of a contrary force which restrains them 
to, and detains them in their orbits, which I therefore call centripetal, would 
tiy off in right lines, with an uniform motion. A projectile, if it was not 
for the force of gravity, would not deviate towards the earth, tut would 



OF NATUJIAL PHILOSOPHY. 7fl 

go off from it in a right line, and that with an uniform motion,, if the re 
sistance of the air was taken away. It is by its gravity that it is drawn 
aside perpetually from its rectilinear course, and made to deviate towards 
the earth, more or less, according to the force of its gravity, and the velo 
city of its motion. The less its gravity is, for the quantity of its matter, 
or the greater the velocity with which it is projected, the less will it devi 
ate from a rectilinear course, and the farther it will go. If a leaden balJ, 
projected from the top of a mountain by the force of gunpowder with a 
given velocity, and in a direction parallel to the horizon, is carried in a 
curve line to the distance of two miles before it falls to the ground ; the 
same, if the resistance of the air were taken away, with a double or decuple 
velocity, would fly twice or ten times as far. And by increasing the velo 
city, we may at pleasure increase the distance to which it might be pro 
jected, and diminish the curvature of the line, which it might describe, till 
at last it should fall at the distance of 10, 30, or 90 degrees, or even might 
go quite round the whole earth before it falls ; or lastly, so that it might 
never fall to the earth, but go forward into the celestial spaces, and pro 
ceed in its motion in iiifiuitum. And after the same manner that a pro 
jectile, by the force of gravity, may be made to revolve in an orbit, and go 
round the whole earth, the moon also, either by the force of gravity, if it 
is endued with gravity, or by any other force, that impels it towards the 
earth, may be perpetually drawn aside towards the earth, out of the r&ti- 
linear way, which by its innate force it would pursue; and would be made 
to revolve in the orbit which it now describes ; nor could the moon with 
out some such force, be retained in its orbit. If this force was too small, 
it would not sufficiently turn the moon out of a rectilinear course : if it 
was too great, it would turn it too much, arid draw down the moon from 
its orbit towards the earth. It is necessary, that the force be of a just 
quantity, and it belongs to the mathematicians to find the force, that may 
serve exactly to retain a body in a given orbit, with a given velocity ; and 
vice versa, to determine the curvilinear way, into which a body projected 
from a given place, with a given velocity, may be made to deviate from 
its natural rectilinear way, by means of a given force. 

The quantity of any centripetal force may be considered as of three 
kinds; aboolu e, accelerative, and motive. 

DEFINITION VI. 

The absolute quantity of a centripetal force is the measure f >f the same 
proportional to the efficacy of the cause that propagates it from the cen 
tre, through the spaces round about. 
Thus the magnetic force is greater in one load-stone and less in another 

according to their sizes and strength of intensity. 



76 THE MATHEMATICAL PRINCIPLES 

DEFINITION VII. 

The accelerative quantity of a centripetal force is the measure, of tht 
same, proportional to the velocity which it generates in a given time. 

Thus the force of the same load-stone is greater at a less distance, and 
less at a greater : also the force of gravity is greater in valleys, less on 
tops of exceeding high mountains ; and yet less (as shall hereafter be shown), 
at greater distances from the body of the earth ; but at equal distan 
ces, it is the same everywhere ; because (taking away, or allowing for, the 
resistance of the air), it equally accelerates all falling bodies, whether heavy 
or light, great or small. 

DEFINITION VIII. 

TJie motive quantity of a centripetal force, is the measure of the samt\ 

proportional to the motion which it generates in a given twip. 

Thus the weight is greater in a greater body, less in a less body ; and. 
in the same body, it is greater near to the earth, and less at remoter dis 
tances. This sort of quantity is the centripetency, or propension of the 
whole body towards the centre, or, as I may say, its weight ; and it is al 
ways known by the quantity of an equal and contrary force just sufficient 
to Ifinder the descent of the body. 

These quantities of forces, we may, for brevity s sake, call by the names 
of motive, accelerative, and absolute forces ; and, for distinction s sake, con 
sider them, with respect to the bodies that tend to the centre ; to the places 
of those bodies ; and to the centre of force towards which they tend ; that 
is to say, I refer the motive force to the body as an endeavour and propen 
sity of the whole towards a centre, arising from the propensities of the 
several parts taken together ; the accelerative force to the place of the 
body, as a certain power or energy diffused from the centre to all places 
around to move the bodies that are in them : and the absolute force to 
the centre, as endued with some cause, without which those motive forces 
would not be propagated through the spaces round about ; whether that 
cause be some central body (siuh as is the load-stone, in the centre of the 
magnetic force, or the earth in the centre of the gravitating force), or 
anything else that does not yet appear. For I here design only to give a 
mathematical notion of those forces, without considering their physical 
causes and seats. 

Wherefore the accelerative force will stand in the same relation to the 
motive, as celerity does to motion. For the quantity of motion arises from 
the celerity drawn into the quantity of matter : and the motive force arises 
from the accelerative force drawn into the same quantity of matter. For 
the sum of the actions of the accelerative force, upon the several ; articles 
of the body, is the motive force of the whole. Hence it is, that near the 



OF NATURAL PHILOSOPHY. 77 

surface of the earth, where the accelerative gravity, or force productive of 
gravity, in all bodies is the same, the motive gravity or the weight is as 
the body : but if we should ascend to higher regions, where the accelerative 
gravity is less, the weight would be equally diminished, and would always 
be as the product of the body, by the accelerative gravity. So in those re 
gions, where the accelerative gravity is diminished into one half, the weight 
of a body two or three times less, will be four or six times less. 

I likewise call attractions and impulses, in the same sense, accelerative, 
and motive ; and use the words attraction, impulse or propensity of any 
sort towards a centre, promiscuously, and indifferently, one for another ; 
considering those forces not physically, but mathematically : wherefore, the 
reader is not to imagine, that by those words, I anywhere take upon me to 
define the kind, or the manner of any action, the causes or the physical 
reason thereof, or that I attribute forces, in a true and physical sense, to 
certain centres (which are only mathematical points) ; when at any time I 
happen to speak of centres as attracting, or as endued with attractive 
powers. 

SCHOLIUM. 

Hitherto I have laid down the definitions of such words as are less 
known, and explained the sense in which I would have them to be under 
stood in the following discourse. I do not define time, space, place and 
motion, as being well known to all. Only I must observe, that the vulgar 
conceive those quantities under no other notions but from the relation they 
bear to sensible objects. And thence arise certain prejudices, for the re 
moving of which, it will be convenient to distinguish them into absolute 
and relative, true and apparent, mathematical and common. 

I. Absolute, true, and mathematical time, of itself, and from its own na 
ture flows equably without regard to anything external, and by another 
name is called duration : relative, apparent, and common time, is some sen 
sible and external (whether accurate or unequable) measure of duration by 
the means of motion, which is commonly used instead of true time ; such 
as an hour, a day, a month, a year. 

II. Absolute space, in its own nature, without regard to anything exter 
nal, remains always similar and immovable. Relative space is some mo 
vable dimension or measure of the absolute spaces ; which our senses de 
termine by its position to bodies ; and which is vulgarly taken for immo 
vable space ; such is the dimension of a subterraneous, an aereal, or celestial 
space, determined by its position in respect of the earth. Absolute and 
relative space, are the same in figure and magnitude ; but they do not re 
main always numerically the same. For if the earth, for instance, moves, 
a space of our air, which relatively and in respect of the earth remains al 
ways the same, will at one time be one part of the absolute space into which 



TS THE MATHEMATICAL PRINCIPLES 

the air passes ; at another time it will be another part of the same, and so. 
absolutely understood, it will be perpetually mutable. 

III. Place is a part of space which a body takes up, and is according to 
the space, either absolute or relative. I say, a part of space ; not the situation, 
nor the external surface of the body. For the places of equal solids are 
always equal ; but their superfices, by reason of their dissimilar figures, are 
often unequal. Positions properly have no quantity, nor are they so much 
the places themselves, as the properties of places. The motion of the whole 
is the same thing with the sum of the motions of the parts ; that is, the 
translation of the whole, out of its place, is the same thing with the sum 
of the translations of the parts out of their places ; and therefore the place 
of the whole is the same thing with the sum of the places of the parts, and 
for that reason, it is internal, and in the whole body. 

IV. Absolute motion is the translation of a body from one absolute 
place into another ; and relative motion, the translation from one relative 
place into another. Thus in a ship under sail, the relative place of a body 
is that part of the ship which the body possesses ; or that part of its cavity 
which the body fills, and which therefore moves together with the ship : 
and relative rest is the continuance of the body in the same part of the 
ship, or of its cavity. But real, absolute rest, is the continuance of the 
body in the same part of that immovable space, in which the ship itself, 
its cavity, and all that it contains, is moved. Wherefore, if the earth is 
really at rest, the body, which relatively rests in the ship, will really and 
absolutely move with the same velocity which the ship has on the earth. 
But if the earth also moves, the true and absolute motion of the body will 
arise, partly from the true motion of the earth, in immovable space ; partly 
from the relative motion of the ship on the earth ; and if the body moves 
also relatively in the ship ; its true motion will arise, partly from the true 
motion of the earth, in immovable space, and partly from the relative mo 
tions as well of the ship on the earth, as of the body in the ship ; and from 
these relative motions will arise the relative motion of the body on the 
earth. As if that part of the earth, where the ship is, was truly moved 
toward the east, with a velocity of 10010 parts; while the ship itself, with 
a fresh gale, and full sails, is carried towards the west, with a velocity ex 
pressed by 10 of those parts ; but a sailor walks in the ship towards the 
east, with 1 part of the said velocity ; then the sailor will be moved truly 
in immovable space towards the east, with a velocity of 10001 parts, and 
relatively on the earth towards the west, with a velocity of 9 of those parts. 

Absolute time, in astronomy, is distinguished from relative, by the equa 
tion or correction of the vulgar time. For the natural days are tr^y un 
equal, though they are commonly considered as equal, and used for a meas 
ure of time ; astronomers correct this inequality for their more accurate 
deducing of the celestial motions. It may be, that there is no such thing 
as an equable motion, whereby time may H accurately measured. All mo 



OF NATURAL PHILOSOPHY. 79 

tions may be accelerated and retarded; but the true, or equable, progress of 
absolute time is liable to no change. The duration or perseverance of the 
existence of things remains the same, whether the motions are swift or slow, 
or none at all : and therefore it ought to be distinguished from what are 
only sensible measures thereof ; and out of which we collect it, by means 
of the astronomical equation. The necessity of which equation, for deter 
mining the times of a phamomenon, is evinced as well from the experiments 
of the pendulum clock, as by eclipses of the satellites of Jupiter. 

As the order of the parts of time is immutable, so also is the order of 
the parts of space. Suppose those parts to be moved out of their places, and 
they will be moved (if the expression may be allowed) out of themselves. 
For times and spaces are, as it were, the places as well of themselves as of 
all other things. All things are placed in time as to order of succession ; 
and in space as _to order of situation. It is from their essence or nature 
that they are places ; and that the primary places of things should be 
moveable, is absurd. These are therefore the absolute places ; and trans 
lations out of those places, are the only absolute motions. 

But because the parts of space cannot be seen, or distinguished from one 
another by our senses, therefore in their stead we use sensible measures of 
them. For from the positions and distances of things from any body con 
sidered as immovable, we define all places ; and then with respect to such 
places, we estimate all motions, considering bodies as transferred from some 
of those places into others. And so, instead of absolute places and motions, 
we use relative ones; and that without any inconvenience in common af 
fairs ; but in philosophical disquisitions, we ought to abstract from our 
senses, and consider things themselves, distinct from what are only sensible 
measures of them. For it may be that there is no body really at rest, to 
which the places and motions of others may be referred. 

But we may distinguish rest and motion, absolute and relative, one from 
the other by their properties, causes and effects. It is a property of rest, 
that bodies really at rest do rest in respect to one another. And therefore 
as it is possible, that in the remote regions of the fixed stars, or perhaps 
far beyond them, there may be some body absolutely at rest ; but impossi 
ble to know, from the position of bodies to one another in our regions 
whether any of these do keep the same position to that remote body; it 
follows that absolute rest cannot be determined from the position of bodies 
in our regions. 

It is a property of motion, that the parts, which retain given positions 
to their wholes, do partake of the motions of those wholes. For all the 
parts of revolving bodies endeavour to recede from the axis of motion ; 
and the impetus of bodies moving forward, arises from the joint impetus 
of all the parts. Therefore, if surrounding bodies are moved, those that 
are relatively at rest within them, will partake of their motion. Upon 
which account, the true and absolute motion of a body cannot be Jeter- 



8C THE MATHEMATICAL PRINCIPLES 

mined by the translation of it from those which only seem to rest ; for the 
external bodies ought not only to appear at rest, but to be really at rest. 
For otherwise, all included bodies, beside their translation from near the 
surrounding ones, partake likewise of their true motions ; and though that 
translation were not made they would not be really at rest, but only seem 
to be so. For the surrounding bodies stand in the like relation to the 
surrounded as the exterior part of a whole does to the interior, or as the 
shell does to the kernel ; but, if the shell moves, the kernel will also 
move, as being part of the whole, without any removal from near the shell. 

A property, near akin to the preceding, is this, that if a place is moved, 
whatever is placed therein moves along with it ; and therefore a body, 
which is moved from a place in motion, partakes also of the motion of its 
place. Upon which account, all motions, from places in motion, are no 
other than parts of entire and absolute motions ; and every entire motion 
is composed of the motion of the body out of its first place, and the 
motion of this place out of its place ; and so on, until we come to some 
immovable place, as in the before-mentioned example of the sailor. Where 
fore, entire and absolute motions can be no otherwise determined than by 
immovable places : and for that reason I did before refer those absolute 
motions to immovable places, but relative ones to movable places. Now 
no other places are immovable but those that, from infinity to infinity, do 
all retain the same given position one to another ; and upon this account 
must ever remain unmoved ; and do thereby constitute immovable space. 

The causes by which true and relative motions are distinguished, one 
from the other, are the forces impressed upon bodies to generate motion. 
True motion is neither generated nor altered, but by some force impressed 
upon the body moved : but relative motion may be generated or altered 
without any force impressed upon the body. For it is sufficient only to 
impress some force on other bodies with which the former is compared, 
that by their giving way, that relation may be changed, in which the re 
lative rest or motion of this other body did consist. Again, true motion 
suffers always some change from any force impressed upon the moving 
body ; but relative motion docs not necessarily undergo any change by such 
forces. For if the same forces are likewise impressed on those other bodies, 
with which the comparison is made, that the relative position may be pre 
served, then that condition will be preserved in which the relative motion 
consists. And therefore any relative motion may be changed when the 
true motion remains unaltered, and the relative may be preserved when the 
true suffers some change. Upon which accounts; true motion does by no 
means consist in such relations. 

The effects whicli distinguish absolute from relative motion arc, the 
forces of receding from the axis of circular motion. For there are no such 
forces in a circular motion purely relative, but in a true and absolute cir 
cular motion., they are greater or less, according t the quantity of the 



OF NATURAL PHILOSOPHY. 1 

motion. If a vessel, hung: by & }ong cord, is so often turned ubout that the 
cord is strongly twisted, then filled with water, and held at rest together 
with the water ; after, by the sudden action of another force, it is whirled 
about the contrary way, and while the cord is untwisting itself, the vessel 
continues for some time in this motion ; the surface of the water will at 
first be plain, as before the vessel began to move : but the vessel; by grad 
ually communicating its motion to the water, will make it begin sensibly 
^to revolve, and recede by little and little from the middle, and ascend to the 
sides of the vessel, forming itself into a concave figure (as I have experi 
enced), and the swifter the motion becomes, the higher will the water rise, 
till at last, performing its revolutions in the same times with the vessel, 
it becomes relatively at rest in it. This ascent of the water shows its en 
deavour to recede from the axis of its motion ; and the true and absolute 
circular motion of the water, which is here directly contrary to the rela- 
tivej discovers itself, and may be measured by this endeavour. At first, 
when the relative motion of the water in the vessel was greatest, it pro 
duced no endeavour to recede from the axis ; the water showed no tendency 
to the circumference, nor any ascent towards the sides of the vessel, but 
remained of a plain surface, and therefore its true circular motion had not 
yet begun. But afterwards, when the relative motion of the water had 
decreased, the ascent thereof towards the sides of the vessel proved its en 
deavour to recede from the axis ; and this endeavour showed the real cir 
cular motion of the water perpetually increasing, till it had acquired its 
greatest quantity, wh en the water rested relatively in the vessel. And 
therefore this endeavour does not depend upon any translation of the water 
in respect of the ambient bodies, nor can true circular motion be defined 
by such translation. There is only one real circular motion of any one 
revolving body, corresponding to only one power of endeavouring to recede 
from its axis of motion, as its proper and adequate effect ; but relative 
motions, in one and the same body, are innumerable, according to the various 
relations it bears to external bodies, and like other relations, arc altogether 
destitute of any real effect, any otherwise than they may perhaps par 
take of that one only true motion. And therefore in their system who 
suppose that our heavens, revolving below the sphere of the fixed stars, 
carry the planets along with them ; the several parts of those heavens, and 
the planets, which are indeed relatively at rest in their heavens, do yet 
really move. For they change their position one to another (which never 
happens to bodies truly at rest), and being carried together with their 
heavens, partake of their motions, and as parts of revolving wholes, 
endeavour to recede from the axis of their motions. 

Wherefore relative quantities are not the quantities themselves, whose 
names they bear, but those sensible measures of them (either accurate cr 
inaccurate), which arc commonly used instead of the measured quantities 
themselves. And if the meaning of words is to he determined bv their 



82 THE MATHEMATICAL PRINCIPLES 

use, then by the names time, space, place and motion, their measures arv 
properly to be understood ; and the expression will be unusual, and purely 
mathematical, if the measured quantities themselves are meant. Upon 
which account, they do strain the sacred writings, who there interpret 
those words for the measured quantities. Nor do those less defile the 
purity of mathematical and philosophical truths, who confound real quan 
tities themselves with their relations and vulgar measures. 

It is indeed a matter of great difficulty to discover, and effectually to 
distinguish, the true motions of particular bodies from the apparent ; be 
cause the parts of that immovable space, in which those motions are per 
formed, do by no means come under the observation of our senses. Yet 
the thing is not altogether desperate : for we have some arguments to 
guide us, partly from the apparent motions, which are the differences of 
the true motions ; partly from the forces, which are the causes and effects 
of the true motions. For instance, if tAvo globes, kept at a given distance 
one from the other by means of a cord that connects them, were revolved 
about their common centre of gravity, we might, from the tension of the 
cord, discover the endeavour of the globes to recede from the axis of their 
motion, and from thence we might compute the quantity of their circular 
motions. And then if any equal forces should be impressed at once on the 
alternate faces of the globes to augment or diminish their circular motions, 
from the increase or decr ase of the tensicn of 1 le cord, we might infer 
the increment or decrement of their motions : and thence would be found 
on what faces those forces ought to be impressed, that the motions of the 
globes might be most augmented ; that is, we might discover their hinder- 
most faces, or those which, in the circular motion, do follow. But the 
faces which follow being known, and consequently the opposite ones that 
precede, we should likewise know the determination of their motions. And 
thus we might find both the quantity and the determination of this circu 
lar motion, even in an immense vacuum, where there was nothing external 
or sensible with which the globes could be compared. But now, if in that 
space some remote bodies were placed that kept always a given position 
one to another, as the fixed stars do in our regions, we could not indeed 
determine from the relative translation of the globes among those bodies, 
whether the motion did belong to the globes or to the bodies. But if we 
observed the cord, and found that its tension was that very tension which 
the motions of the globes required, we might conclude the motion to be in 
the globes, and the bodies to be at rest ; and then, lastly, from the trans 
lation of the globes among the bodies, we should find the determination oi 
their motions. But how we are to collect the true motions from their 
causes, effects, and apparent differences ; and, vice versa, how from the mo 
tions, either true or apparent, we may come to the knowledge of theii 
causes and effects, shall be explained more at large in the following tra<;t 
For to this end it was that I composed it. 



OF NATURAL PHILOSOPHY. 



AXIOMS, OR LAWS OF MOTION. 

LAW I. 

Hvery body perseveres in its state of rest, or of uniform motion in a ri^ht 
line, unless it is compelled to change that state by forces impressed 
thereon. 

PROJECTILES persevere in their motions, so far as they are not retarded 
by the resistance of the air, or impelled downwards by the force of gravity 
A top, whose parts by their cohesion are perpetually drawn aside from 
rectilinear motions, does not cease its rotation, otherwise than as it is re 
tarded by the air. The greater bodies of the planets and comets, meeting 
with less resistance in more free spaces, preserve then jDotions both pro 
gressive and circular for a much longer time. 

LAW II. 

The alteration of motion is ever proportional to the motive force imp reus 
ed ; and is made in the direction of the right line in. which that force 
is impressed. 

If any force generates a motion, a double force will generate double the 
motion, a triple force triple the motion, whether that force be impressed 
altogether and at once, or gradually and successively. And this motion 
(being always directed the same way with the generating force), if the body 
moved before, is added to or subducted from the former motion, according 
as they directly conspire with or are directly contrary to each other ; or 
obliquely joined, when they are oblique, so as to produce a new motion 
compounded from the determination of both. 

LAW III. 

To every action there is always opposed an equal reaction : or the mu 
tual actions of two bodies upon each other are always equal, and di 
rected to contrary parts. 

Whatever draws or presses another is as much drawn or pressed by that 
other. If you press a stone with your finger, the finger is also pressed by 
the stone. If a horse draws a stone tied to a rope, the horse (if I may so 
say) will be equally drawn back towards the stone: for the distended rope, 
by the same endeavour to relax or unbend itself, will draw the horse as 
much towards the stone, as it does the stone towards the horse, and will 
obstruct the progress of the one as much as it advances that of the other. 



84 THE MATHEMATICAL PRINCIPLES 

If a body impinge upon another, and by its force change the motion of (It* 
other, that body also (because of the equality of the mutual pressure) will 
undergo an equal change, in its own motion, towards the contrary part. 
The changes made by these actions are equal, not in the velocities but in 
the motions of bodies ; that is to say, if the bodies are not hindered by any 
other impediments. For, because the motions are equally changed, the 
changes of the velocities made towards contrary parts are reciprocally pro 
portional to the bodies. This law takes place also in attractions, as will 
be proved in the next scholium. 

COROLLARY I. 

A body by two forces conjoined will describe the diagonal of a parallelo 
gram, in the same time that it wovld describe the sides, by those forces 
apart. 

If a body in a given time, by the force M impressed 
apart in the place A, should with an uniform motion / 
be carried from A to B ; and by the force N impressed 
apart in the same place, should be carried from A to c ~\) 
C ; complete the parallelogram ABCD, and, by both forces acting together, 
it will in the same time be carried in the diagonal from A to D. For 
since the force N acts in the direction of the line AC, parallel to BD, this 
force (by the second law) will not at all alter the velocity generated by the 
other force M, by which the body is carried towards the line BD. The 
body therefore will arrive at the line BD in the same time, whether the 
rorce N be impressed or not ; and therefore at the end of that time it will 
he found somewhere in the line BD. By the same argument, at the end 
of the same time it AY ill be found somewhere in the line CD. Therefore it 
will be found in the point D, where both lines meet. But it will move in 
;i right line from A to D, by Law I. 

COROLLARY II. 

And hence is explained the composition of any one direct force AD, out 
of any two oblique forces AC and CD ; and, on the contrary, the re 
solution of any one direct force AD into two oblique forces AC and 
CD : which composition and resolution are abundantly confirmed from, 
mechanics. 

As if the unequal radii OM and ON drawn from the centre O of any 
wheel, should sustain the weights A and P by the cords MA and NP ; and 
the forces of those weights to move the wheel were required. Through the 
rentre O draw the right line KOL, meeting the cords perpendicularly in 
A and L; and from the centre O, with OL the greater of the distances 




OF NATURAL PHILOSOPHY. 

OK arid OL, describe a circle, meeting the cord 
MA in D : and drawing OD, make AC paral- "^ 
lei and DC perpendicular thereto. Now, it 
being indifferent whether the points K, L, D, of 
the cords be lixed to the plane of the wheel or 
not, the weights will have the same effect 
whether they are suspended from the points K 
and L, or from D and L. Let the whole force 
of the weight A be represented by the line AD, 
and let it be resolved into the forces AC and 
CD ; of which the force AC, drawing the radius 
OD directly from the centre, will have no effect to move the wheel : but 
the other force DC, drawing the radius DO perpendicularly, will have the 
same effect as if it drew perpendicularly the radius OL equal to OD ; that 
is, it w ill have the same effect as the weight P, if that weight is to the 
weight A as the force DC is to the force DA ; that is (because of the sim 
ilar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A 
and P, which are reciprocally as the radii OK and OL that lie in the same 
right line, will be equipollent, and so remain in equilibrio ; which is the well 
known property of the balance, the lever, and the wheel. If either weight is 
greater than in this ratio, its force to move the wheel will be so much greater. 

If the weight p, equal to the weight P, is partly suspended by the 
cord NJO, partly sustained by the oblique plane pG ; draw p}i, NH, the 
former perpendicular to the horizon, the latter to the plane pG ; and if 
the force of the weight p tending downwards is represented by the line 
/?H, it may be resolved into the forces joN, HN. If there was any plane 
/?Q, perpendicular to the cord y?N, cutting the other plane pG in a line 
parallel to the horizon, and the weight p was supported only by those 
planes pQ, pG, it would press those planes perpendicularly with the forces 
pN, HN; to wit, the plane joQ, with the force joN, and the plane pG with 
the force HN. And therefore if the plane pQ was taken away, so thnt 
the weight might stretch the cord, because the cord, now sustaining the 
weight, supplies the place of the plane that was removed, it will be strained 
by the same force joN which pressed upon the plane before. Therefore, 
the tension of this oblique cord joN will be to that of the other perpendic 
ular cord PN as jt?N to joH. And therefore if the weight p is to the 
weight A in a ratio compounded of the reciprocal ratio of the least distances 
of the cords PN, AM, from the centre of the wheel, and of the direct ratio of 
pH tojoN, the weights will have the same effect towards moving the wheel, 
and will therefore sustain each other : as any one may find by experiment. 

But the weight p pressing upon those two oblique planes, may be con 
sidered as a wedge between the two internal surfaces of a body split by it; 
and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi 



8G THE MATHEMATICAL PRINCIPLES 

because the force with which the weight p presses the plane pQi is to the 
force with which the same, whether by its own gravity, or by the blow of 
a mallet, is impelled in the direction of the line joH towards both the 
planes, as joN to pH ; and to the force with which it presses the other 
plane pG, as joN to NH. And thus the force of the screw may be deduced 
from a like resolution of forces ; it being no other than a wedge impelled 
with the force of a lever. Therefore the use of this Corollary spreads far 
and wide, and by that diffusive extent the truth thereof is farther con 
firmed. For on what has been said depends the whole doctrine of mechan 
ics variously demonstrated by different authors. For from hence are easily 
deduced the forces of machines, which are compounded of wheels, pullics, 
levers, cords, and weights, ascending directly or obliquely, and other mechan 
ical powers ; as also the force of the tendons to move the bones of animals. 

COROLLARY III. 

The (/uaittity of motion, which is collected by taking the sum of the mo 
tions directed towards the same parts, and the difference of those that 
are directed to contrary parts, suffers no change from the action oj 
bodies among themselves. 

For action and its opposite re-action are equal, by Law III, and there 
fore, by Law II, they produce in the motions equal changes towards oppo 
site parts. Therefore if the motions are directed towards the same parts. 
whatever is added to the motion of the preceding body will be subducted 
from the motion of that which follows ; so that the sum will be the same 
as before. If the bodies meet, with contrary motions, there will be an 
equal deduction from the motions of both ; and therefore the difference of 
the motions directed towards opposite parts will remain the same. 

Thus if a spherical body A with two parts of velocity is triple of a 
spherical body B which follows in the same right line with ten parts of 
velocity, the motion of A will be to that of B as 6 to 10. Suppose, 
then, their motions to be of 6 parts and of 10 parts, and the sum will be 
16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, 
or 5 parts of motion, B will lose as many ; and therefore after reflexion 
A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the 
sum remaining always of 16 parts as before. If the body A acquire 9, 
10, 11, or 12 parts of motion, and therefore after meeting proceed with 
15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, 
will either proceed with 1 part, having lost 9, or stop and remain at rest, 
as having lost its whole progressive motion of 10 parts ; or it will go back 
with 1 part, having not only lost its whole motion, but (if 1 may so say) 
one part more; or it will go back with 2 parts, because a progressive mo 
tion of 12 parts is taken off. And so the sums of the Conspiring motions 
15 ,1, or 16-1-0, and the differences of the contrary i otions 17 1 and 



OF NATURAL PHILOSOPHY. 

[S 2, will always be equal to 16 parts, as they were before tie meeting 
and reflexion of the bodies. But, the motions being known with whicli 
the bodies proceed after reflexion, the velocity of either will be also known, 
by taking the velocity after to the velocity before reflexion, as the motion 
after is to the motion before. As in the last case, where the motion of tho 
body A was of parts before reflexion and of IS parts after, and the 
velocity was of 2 parts before reflexion, the velocity thereof after reflexion 
will be found to be of 6 parts ; by saying, as the parts of motion before 
to 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after. 
But if the bodies are cither not spherical, or, moving in different right 
lines, impinge obliquely one upon the other, and their mot ons after re 
flexion are required, in those cases we are first to determine the position 
of the plane that touches the concurring bodies in the point of concourse , 
then the motion of each body (by Corol. II) is to be resolved into two, one 
perpendicular to that plane, and the other parallel to it. This done, be 
cause the bodies act upon each other in the direction of a line perpendicu 
lar to this plane, the parallel motions are to be retained the same after 
reflexion as before ; and to the perpendicular motions we are to assign 
equal changes towards the contrary parts ; in such manner that the sum 
of the conspiring and the difference of the contrary motions may remain 
the same as before. From such kind of reflexions also sometimes arise 
the circular motions of bodies about their own centres. But these are 
cases which I do not consider in what follows ; and it would be too tedious 
to demonstrate every particular that relates to this subject. 

COROLLARY IV. 

The common centre of gravity of two or more bodies does not alter its 
state of motion or rest by the actions of the bodies among themselves ; 
and therefore the common centre of gravity of all bodies acting upon 
each other (excluding outward actions and impediments) is either at 
rest, or moves uniformly in a right line. 

For if two points proceed with an uniform motion in right lines, and 
their distance be divided in a given ratio, the dividing point will be either 
at rest, or proceed uniformly in a right line. This is demonstrated here 
after in Lem. XXIII and its Corol., when the points are moved in the same 
plane ; and by a like way of arguing, it may be demonstrated when the 
points are not moved in the same plane. Therefore if any number of 
Kdies move uniformly in right lines, the common centre of gravity of any 
two of them is either at rest, or proceeds uniformly in a right line ; because 
the line which connects the centres of those two bodies so moving is divided at 
that common centre in a given ratio. In like manner the common centre 
of those two and that of a third body will be either at rest or moving uni 
formly in aright line because at that centre the distance 1 etween th? 



88 THE MATHEMATICAL PRINCIPLES 

common centre of the two bodies, and the centre of this last, is divided in 
a given ratio. In like manner the common centre of these three, and of a 
fourth body, is either at rest, or moves uniformly in a right line ; because 
the distance between the common centre of the three bodies, and the centre 
of the fourth is there also divided in a given ratio, and so on m itifinitum. 
Therefore, in a system of bodies where there is neither any mutual action 
among themselves, nor any foreign force impressed upon them from without, 
and which consequently move uniformly in right lines, the common centre of 
gravity of them all is either at rest or moves uniformly forward in a right line. 
Moreover, in a system of two bodies mutually acting upon each other, 
since the distances between their centres and the common centre of gravity 
of both are reciprocally as the bodies, the relative motions of those bodies, 
whether of approaching to or of receding from that centre, will be equal 
among themselves. Therefore since the changes which happen to motions 
are equal and directed to contrary parts, the common centre of those bodies, 
by their mutual action between themselves, is neither promoted nor re 
tarded, nor suffers any change as to its state of motion or rest. But in a 
system of several bodies, because the common centre of gravity of any two 
acting mutually upon each other suffers no change in its state by that ac 
tion : and much less the common centre of gravity of the others with which 
that action does not intervene ; but the distance between those two centres 
is divided by the common centre of gravity of all the bodies into parts re 
ciprocally proportional to the total sums of those bodies whose centres they 
are : and therefore while those two centres retain their state of motion or 
rest, xhe common centre of all does also retain its state : it is manifest that 
the common centre of all never suffers any change in the state of its mo 
tion or rest from the actions of any two bodies between themselves. But 
in such & system all the actions of the bodies among themselves either hap 
pen between two bodies, or are composed of actions interchanged between 
some two bodies ; and therefore they do never produce any alteration in 
the comrrv n centre of alias to its state of motion or rest. Wherefore 
tiince that centre, when the bodies do not act mutually one upon another, 
Oilier is nt rest or moves uniformly forward in some right line, it will, 
:v\>U7ithst?nding the mutual actions of the bodies among themselves, always 
jAY-jevere in its state, either of rest, or of proceeding uniformly in a right 
liiv,, unless it is forced out of this state by the action of some power im- 
prev^-d from without upon the whole system. And therefore the same law 
take* 1 place in a system consisting of many bodies as in one single body, 
with wsgard to their persevering in their state of motion or of rest. For 
the pi \\jressive motion, whether of one single body, or of a whole system of 
bodies us always to be estimated from the motion of the centre of gravity. 

COROLLARY V. 

The motions cf bcdies included in a given space a ~e Ike same among 



OF NATURAL PHILOSOPHY. 89 

themselves, whether that space is at rest, or moves uniformly forwards 

in a right line without any circular motion. 

For the differences of the motions tending towards the same parts, and 
the sums of those that tend towards contrary parts, are, at first (by sup 
position), in both cases the same ; and it is from those sums and differences 
that the collisions and impulses do arise with which the bodies mutually 
impinge one upon another. Wherefore (by Law II), the effects of those 
collisions will be equal in both cases ; and therefore the mutual motions 
of the bodies among themselves in the one case will remain equal to the 
mutual motions of the bodies among themselves in the other. A clear 
proof of which we have from the experiment of a ship ; where all motions 
happen after the same manner, whether the ship is at rest, or is carried 
uniformly forwards in a right line. 

COROLLARY VI. 

If bodies, any how moved among themselves, are urged in the direct-ton 
of parallel lines by equal accelerative forces, they will all continue to 
move among themselves, after the same manner as if they had been 
urged by no such forces. 

For these forces acting equally (with respect to the quantities of the 
DO dies to be moved), and in the direction of parallel lines, will (by Law II) 
move all the bodies equally (as to velocity), and therefore will never pro 
duce any change in the positions or motions of the bodies among themselves. 

SCHOLIUM. 

Hitherto I have laid down such principles as have been received by math 
ematicians, and are confirmed by abundance of experiments. By the first 
two Laws and the first two Corollaries, Galileo discovered that the de 
scent of bodies observed the duplicate ratio of the time, and that the mo 
tion of projectiles was in the curve of a parabola; experience agreeing 
with both, unless so far as these motions are a little retarded by the re 
sistance of the air. When a body is falling, the uniform force of its 
gravity acting equally, impresses, in equal particles of time, equal forces 
upon that body, and therefore generates equal velocities; and in the whole 
time impresses a whole force, and generates a whole velocity proportional 
to the time. And the spaces described in proportional times are as the 
velocities and the times conjunctly ; that is, in a duplicate ratio of the 
times. And when a body is thrown upwards, its uniform gravity im 
presses forces and takes off velocities proportional to the times ; and the 
times of ascending to the greatest heights are as the velocities to be taken 
off, and those heights are as the velocities and the times conjunetly, or ir, 
the duplicate ratio of the velocities. And if a body be projected in any 
direction, the motion arising from its projection jS compounded with the 



90 



THE MATHEMATICAL PRINCIPLES 




motion arising from its gravity. As if the body A by its motion of pio- 
jection alone could describe in a given time the right line 
AB, and with its motion of falling alone could describe in 
the same time the altitude AC ; complete the paralello- 
gram ABDC, and the body by that compounded motion 
will at the end of the time be found in the place D ; and 
the curve line AED, which that body describes, will be a 
parabola, to which the right line AB will be a tangent in 
A ; and whose ordinate BD will be as the square of the line AB. On the 
same Laws and Corollaries depend those things which have been demon 
strated concerning the times of the vibration of pendulums, and are con 
firmed by the daily experiments of pendulum clocks. By the same, to 
gether with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huv- 
gens, the greatest geometers of our times, did severally determine the rules 
of the congress and reflexion of hard bodies, and much about the same 
time communicated their discoveries to the Royal Society, exactly agreeing 
among themselves as to those rules. Dr. Wallis, indeed, was something 
more early in the publication ; then followed Sir Christopher Wren, and, 
lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of 
the thing before the Royal Society by the experiment of pendulums, which 
Mr. Mariottc soon after thought fit to explain in a treatise entirely upon 
that subject. But to bring this experiment to an accurate agreement with 
the theory, we are to have a due regard as well to the resistance of the air 



bodies. 



Let the spherical bodies 
CD F II 




as to the clastic force of the concurrin 
A, B be suspended by the parallel and 
equal strings AC, Bl), from the centres 
C, D. About these centres, with those 
intervals, describe the semicircles EAF, 
GBH, bisected by the radii CA, DB. 
Bring the body A to any point R of the 
arc EAF, and (withdrawing the body 
B) let it go from thence, and after one oscillation suppose it to return to 
the point V : then RV will be the retardation arising from the resistance 
of the air. Of this RV let ST be a fourth part, situated in the middle. 
to wit, so as RS and TV may be equal, and RS may be to ST as 3 to 2 
then will ST represent very nearly the retardation during the descent 
from S to A. Restore the body B to its place: and, supjx sing the body 
A to be let fall from the point S, the velocity thereof in the place of re 
flexion A, without sensible error, will be the same as if it had descended 
m vacit.o from the point T. Upon which account this velocity may be 
represented by the chord of the arc TA. For it is a proposition well 
known to geometers, that the velocity of a pendulous body in the loAvest 
point is as the chord of the arc which it has described in its descent. Aftci 



OF NATUltAL PHILOSOPHY. 9 I 

reflexion, suppose the body A comes to the place s, and the body B to the 
place k. Withdraw the body B, and find the place v, from which if the 
body A, being let go, should after one oscillation return to the place r, st 
may be a fourth part of rv. so placed in the middle thereof as to leave is 
equal to tv, and let the chord of the arc tA represent the velocity which 
the body A had in the place A immediately after reflexion. For t will be 
the true and correct place to which the body A should have ascended, if 
the resistance of the air had been taken off. In the s.ime way we are to 
correct the place k to which the body B ascends, by finding the place I to 
which it should have ascended in vacuo. And thus everything may be 
subjected to experiment, in the same manner as if we were really placed 
in vacuo. These things being done, we are to take the product (if I may 
so say) of the body A, by the chord of the arc TA (which represents its 
velocity), that we may have its motion in the place A immediately before 
reflexion ; and then by the chord of the arc /A, that we may have its mo 
tion in the place A immediately after reflexion. And so we are to take 
the product of the body B by the chord of the arc B/, that we may have 
the motion of the same immediately after reflexion. And in like manner, 
when two bodies are let go together from different places, we are to find 
the motion of each, as well before as after reflexion; and then we may 
compare the motions between themselves, and collect the effects of the re 
flexion. Thus trying the thing with pendulums of ten feet, in unequal 
as well as equal bodies, and making the bodies to concur after a descent 
through large spaces, as of 8, 12, or 16 feet, I found always, without an 
error of 3 inches, that when the bodies concurred together directly, equal 
changes towards the contrary parts were produced in their motions, and, 
of consequence, that the action and reaction were always equal. As if the 
body A impinged upon the body B at rest with 9 parts of motion, and 
losing 7, proceeded after reflexion with 2, the body B was carried back 
wards with those 7 parts. If the bodies concurred with contrary motions, 
A with twelve parts of motion, and B with six, then if A receded with J4, 
B receded with 8 ; to wit, with a deduction of 14 parts of motion on 
each side. For from the motion of A subducting twelve parts, nothing 
will remain ; but subducting 2 parts more, a motion will be generated of 
2 parts towards the contrary way ; and so, from the motion of the body 
B of 6 parts, subducting 14 parts, a motion is generated of 8 parts towards 
the contrary way. But if the bodies were made both to move towards the 
same way, A, the swifter, with 14 parts of motion, B, the slower, with 5, 
and after reflexion A went on with 5, B likewise went on with 14 parts ; 
9 parts being transferred from A to B. And so in other cases. By the 
congress and collision of bodies, the quantity of motion, collected from the 
sum of the motions directed towards the same way, or from the difference, 
of those that were directed towards contrary ways, was never changed. 
For the error of an inch or two in measures may be easily ascribed to tht 



92 THE MATHEMATICAL PRINCIPLES 

difficulty of executing everything with accuracy. It was not easy to let 
go the two pendulums so exactly together that the bodies should impinge 
one upon the other in the lowermost place AB ; nor to mark the places s, 
and k y to which the bodies ascended after congress. Nay, and some errors, 
too, might have happened from the unequal density of the parts of the pen 
dulous bodies themselves, and from the irregularity of the texture pro 
ceeding from other causes. 

But to prevent an objection that may perhaps be alledged against the 
rule, for the proof of which this experiment was made, as if this rule did 
suppose that the bodies were either absolutely hard, or at least perfectly 
elastic (whereas no such bodies are to be found in nature), 1 must add. that 
the experiments we have been describing, by no means depending upon 
that quality of hardness, do succeed as well in soft as in hard bodies. For 
if the rule is to be tried in bodies not perfectly hard, we are only to di 
minish the reflexion in such a certain proportion as the quantity of the 
elastic force requires. By the theory of Wren and Huygens, bodies abso 
lutely hard return one from another with the same velocity with which 
they meet. But this may be affirmed with more certainty of bodies per 
fectly elastic. In bodies imperfectly elastic the velocity of the return is to 
be diminished together with the elastic force ; because that force (except 
when the parts of bodies are bruised by their congress, or suffer some such 
extension as happens under the strokes of a hammer) is (as far as I can per 
ceive) certain and determined, and makes the bodies to return one from 
the other with a relative velocity, which is in a given ratio to that relative 
velocity with which they met. This I tried in balls of wool, made up 
tightly, and strongly compressed. For, first, by letting go the pendulous 
bodies, and measuring their reflexion, I determined the quantity of their 
elastic force ; and then, according to this force, estimated the reflexions 
that ought to happen in other cases of congress. And with this computa 
tion other experiments made afterwards did accordingly agree ; the balls 
always receding one from the other with a relative velocity, which was to 
the relative velocity with which they met as about 5 to 9. Balls of steel 
returned with almost the same velocity : those of cork with a velocity some-^ 
thing less ; but in balls of glass the proportion was as about 15 to 16. 
And thus the third Law, so far as it regards percussions and reflexions, is 
proved by a theory exactly agreeing with experience. 

In attractions, I briefly demonstrate the thing after this manner. Sup 
pose an obstacle is interposed to hinder the congress of any two bodies A. 
B, mutually attracting one the other : then if either body, as A, is more 
attracted towards the other body B, than that other body B is towards the 
first body A, the obstacle will be more strongly urged by the pressure of 
the body A than by the pressure of the body B, and therefore will not 
remain in equilibrio : but the stronger pressure will prevail, and will make 
the system of the two bodies, together with the obstacle, to move directly 



OF NATURAL PHILOSOPHY. 



93 



towards the parts on which B lies ; arid in free spaces, to go forward in 
infmitiim with a motion perpetually accelerated ; which is absurd and 
contrary to the first Law. For, by the first Law, the system ought to per 
severe in its state of rest, or of moving uniformly forward in a right line : 
and therefore the bodies must equally press the obstacle, and be equally 
attracted one by the other. I made the experiment on the loadstone and 
iron. If these, placed apart in proper vessels, are made to float by one 
another in standing water, neither of them will propel the other ; but, 
by being equally attracted, they will sustain each other s pressure, and rest 
at last in an equilibrium. 

So the gravitation betwixt the earth and its parts is mutual. Let the 
earth FI be cut by any plane EG into two parts EGF 
and EGI, and their weights one towards the other 
will be mutually equal. For if by another plane 
HK, parallel to the former EG, the greater partFJ 
EGI is cut into two parts EGKH and HKI. 
whereof HKI is equal to the part EFG, first cut 
oft , it is evident that the middle part EGKH, will 
have no propension by its proper weight towards either side, but will hang 
as it were, and rest in an equilibrium betwixt both. But the one extreme 
part HKI will with its whole weight bear upon and press the middle part 
towards the other extreme part EGF : and therefore the force with which 
EGI, the sum of the parts HKI and EGKH, tends towards the third part 
EGF, is equal to the weight of the part HKI, that is, to the weight of 
the third part EGF. And therefore the weights of the two parts EGI 
and EGF, one towards the other, are equal, as I was to prove. And in 
deed if those weights were not equal, the whole earth floating in the non- 
resisting aether would give way to the greater weight, and, retiring from 
it, would be carried off in infinitum. 

And as those bodies are equipollent in the congress and reflexion, whose 
velocities are reciprocally as their innate forces, so in the use of mechanic 
instruments those agents are equipollent, and mutually sustain each the 
contrary pressure of the other, whose velocities, estimated according to the 
determination of the forces, are reciprocally as the forces. 

So those weights are of equal force to move the arms of a balance; 
which during the play of the balance are reciprocally as their velocities 
upw ards and downwards ; that is, if the ascent or descent is direct, those 
weights are of equal force, which are reciprocally as the distances of the 
points at which they are suspended from the axis oi the balance : but if 
they are turned aside by the interposition of oblique planes, or other ob 
stacles, and made to ascend or descend obliquely, those bodies will be 
equipollent, w T hich are reciprocally as the heights of their ascent and de 
scent taken according to the perpendicular ; and that on account of the 
determination of gravity downwards. 



94 THE MATHEMATICAL PRINCIPLES 

And in like manner in the pully, or in a combination of pullies, the 
force of a hand drawing the rope directly, which is to the weight, whethel 
ascending directly or obliquely, as the velocity of the perpendicular ascent 
of the weight to the velocity of the hand that draws the rope, will sustain 
the weight. 

In clocks and such like instruments, made up from a combination of 
wheels, the contrary forces that promote and impede the motion of the 
wheels, if they are reciprocally as the velocities of the parts of the wheel 
on which they are impressed, will mutually sustain the one the other. 

The force of the screw to press a body is to the force of the hand that 
turns the handles by which it is moved as the circular velocity of the 
handle in that part where it is impelled by the hand is to the progressive 
velocity of the screw towards the pressed body. 

The forces by which the wedge presses or drives the two parts of the 
wood it cleaves are to the force of the mallet upon the wedge as the pro- 
press of the wedge in the direction of the force impressed upon it by the 
mallet is to the velocity with which the parts of the wood yield to the 
wedge, in the direction of lines perpendicular to the sides of the wedge. 
And the like account is to be given of all machines. 

The power and use of machines consist only in this, that by diminishing 
the velocity we may augment the force, and the contrary : from whence 
in all sorts of proper machines, we have the solution of this problem ; 7 
move a given weight with a given power, or with a given force to over 
come any other given resistance. For if machines are so contrived that the 
velocities of the agent and resistant are reciprocally as their forces, the 
agent will just sustain the resistant, but with a greater disparity of ve 
locity will overcome it. So that if the disparity of velocities is so great 
as to overcome all that resistance which commonly arises either from the 
attrition of contiguous bodies as they slide by one another, or from the 
cohesion of continuous bodies that are to be separated, or from the weights 
of bodies to be raised, the excess of the force remaining, after all those re 
sistances are overcome, will produce an acceleration of motion proportional 
thereto, as well in the parts of {he machine as in the resisting body. But 
to treat of mechanics is not my present business. I was only willing to 
show by those examples the great extent and certainty of the third Law ot 
motion. For if we estimate the action of the agent from its force and 
velocity conjunctly, and likewise the reaction of the impediment conjuncth 
from the velocities of its several parts, and from the forces of resistance 
arising from the attrition, cohesion, weight, and acceleration of those parts, 
the action and reaction YL the use of all sorts of machines will b" found 
always equal to one another. And so far as the action is propagated by 
the intervening instruments, and at last impressed upon tic resisting 
body, the ultimate determination of the action will be always contrary to 
the determination of the reaction. 



OF NATURAL PHILOSOPHY 



95 



BOOK I. 



OF THE MOTION OF BODIES. 



SECTION I. 

Of the method of first and last ratios of quantities, by the help wJicreoj 
we demonstrate the propositions that follow. 

LEMMA I. 

Quantities, and the ratios of quantities, which in any finite time converge 
continually to equality, and before the end of that time approach nearer 
the one to the other than by any given difference, become ultimately 
equal. 

If you deny it, suppose them to be ultimately unequal, and let D be 
their ultimate difference. Therefore they cannot approach nearer to 
equality than by that given difference D ; which is against the supposition, 

LEMMA II. 

If in any figure AacE, terminated by the right (f 
lines A a. AE, and the curve acE, there be in 
scribed any number of parallelograms Ab, Be, 
Cd, fyc., comprehended under equal bases AB, 
BC, CD, ^c., and the sides, Bb, Cc, Dd, ^c., 
parallel to one side Aa of the figure ; and the 
parallelograms aKbl, bLcm, cMdn, *c., are com 
pleted. Then if the breadth of those parallelo- \ 
grams be supposed to be diminisJied, and their X BF C D |; 
number to be augmented in infinitum : / say, that :he ultimate ratios 
which the inscribed fignre AKbLcMdD, the tin nmscribed figure 
AalbmcndoE, and en rvilijiear figure AabcdE, will have to one another, 
are ratios of equality. 

For the difference of the inscribed and circumscribed figures is the sum 
of the parallelograms K7, Lw, M//. Do. that is (from the equality of all 
their bases), the rectangle under one of their bases K6 and the sum of their 
altitudes Aa, that is, the rectangle ABla. But this rectangle, because 




M 



a 



96 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1 




its breadth AB is supposed diminished in infinitum, becomes less than 
any given space. And therefore (by Lem. I) the figures inscribed and 
circumscribed become ultimately equal one to the other; and much more 
will the intermediate curvilinear figure be ultimately equal to either* 
Q.E.D. 

LEMMA III. 

The same ultimate ratios are also ratios of equality, when the breadth^ 
AB, BC, DC, fyc., of the parallelograms are unequal, and are all di 
minished in infinitum. 

For suppose AF equal to the greatest breadth, and 
complete the parallelogram FAaf. This parallelo 
gram will be greater than the difference of the in 
scribed and circumscribed figures ; but, because its 
breadth AF is diminished in infinitum, it will be 
come less than any given rectangle. Q.E.D. 

COR. 1. Hence the ultimate sum of those evanes 
cent parallelograms will in all parts coincide with 

the curvilinear figure. A BF C D E 

COR. 2. Much more will the rectilinear figure^comprehendcd under tne 
chords of the evanescent arcs ab, be, cd, (fee., ultimately coincide with tl.c 
curvilinear figure. 

COR. 3. And also the circumscribed rectilinear figure comprehended 
under the tangents of the same arcs. 

COR. 4 And therefore these ultimate figures (as to their perimeters acE) 
are not rectilinear, but curvilinear limi s of rectilinear figures. 

LEMMA IV. 

If in two figures AacE, PprT, you inscribe (as before) 

two ranks of parallelograms, an equal number in 

each rank, and, when their breadths are diminished 

in infinitum. the ultimate ratios of the parallelograms 

in one figure to those in the other, each to each respec 
tively, are the same; I say, that those two figures 

AacE, PprT, are to one another in that same ratio. 

For as the parallelograms in the one are severally to p 
the parallelograms in the other, so (by composition) is the < 
sum of all in the one to the sum of all in the other : and 
so is the one figure to the other; because (by Lem. Ill) the 
former figure to the former sum, and the latter figure to the 
latter sum, are both in the ratio of equality. Q.E.D. 

COR. Hence if two quantities of any kind are any 
how divided into an equal number of parts, and those A 







SEC. I.] OF NATURAL PHILOSOPHY. 97 

parts, when their number is augmented, and their magnitude diminished 
in infinitum, have a given ratio one to the other, the first to the first, the 
second to the second, and so on in order, the whole quantities will be one to 
the other in that same given ratio. For if, in the figures of this Lemma, 
the parallelograms are taken one to the other in the ratio of the parts, the 
sum of the parts will always be as the sum of the parallelograms ; and 
therefore supposing the number of the parallelograms and parts to be aug 
mented, and their magnitudes diminished in infinitum, those sums will be 
in the ultimate ratio of the parallelogram in the one figure to the corres 
pondent parallelogram in the other ; that is (by the supposition), in the 
ultimate ratio of any part of the one quantity to the correspondent part of 
the other. 

LEMMA V. 

In similar figures, all sorts of homologous sides, whether curvilinear or 
rectilinear, are proportional ; and the areas are in the duplicate ratio 
of the homologous sides. 

LEMMA VI. 

If any arc ACB, given in position, is snb- _j 
tended by its chord AB, and in any point 
A, in the middle of the contiinied curva 
ture, is touched by a right line AD, pro 
duced both ways ; then if the points A R 
and B approach one another and meet, 
I say, the angle RAT), contained between, 
the chord and the tangent, will be dimin- ? 
ished in infinitum, a/id ultimately will vanish. 

For if that angle does not vanish, the arc ACB will contain with the 
tangent AD an angle equal to a rectilinear angle ; and therefore the cur 
vature at the point A will not be continued, which is against the supposi 
tion. 

LEMMA VII. 

The same things being supposed, I say that the ultimate ratio of the arc, 

chord, and tangent, any one to any other, is the ratio of equality. 

For while the point B approaches towards the point A, consider always 

AB and AD as produced to the remote points b and d, and parallel to the 

secant BD draw bd : and let the arc Acb be always similar to the arc 

ACB. Then, supposing the points A and B to coincide, the angle dAb 

will vanish, by the preceding Lemma; and therefore the right lines Ab, 

Arf (which are always finite), and the intermediate arc Acb, will coincide, 

and become equal among themselves. Wheref ,re, the right lines AB, AD, 





98 THE MATHEMATICAL PRINCIPLES [SEC. I. 

and the intermediate arc ACB (which are always proportional to the 
former), will vanish, and ultimately acquire the ratio of equality. Q.E.D. 

COR. 1. Whence if through B we draw A 

BP parallel to the tangent, always cutting 
any right line AF passing through A in F/ i- 

P, this line BP will be ultimately in the 

ratio of equality with the evanescent arc ACB ; because, completing the 
parallelogram APBD, it is always in a ratio of equality with AD. 

COR. 2. And if through B and A more right lines are drawn, as BE, 
I5D, AF, AG, cutting the tangent AD and its parallel BP : the ultimate 
ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, 
any one to any other, will be the ratio of equality. 

COR. 3. And therefore in all our reasoning about ultimate ratios, we 
may freely use any one of those lines for any other. 

LEMMA VIII. 

If the right lines AR, BR, with the arc ACB, the chord AB, and the 
tangent AD, constitute three triangles RAB. RACB, RAD, and the 
points A and B approach and meet : I say, that the ultimate form oj 
these evanescent triangles is that of similitude, and their ultimate 
ratio that of equality. 

For while the point B approaches towards A 
the point A, consider always AB, AD, AR, 
as produced to the remote points b, d, and r, 
and rbd as drawn parallel to RD, and let 
the arc Acb be always similar to the arc 
ACB. Then supposing the points A and B 
to coincide, the angle bAd will vanish ; and 
therefore the three triangles rAb, rAcb,rAd 
^which are always finite), will coincide, and on that account become both 
similar and equal. And therefore the triangles RAB. RACB, RAD 
which are always similar and proportional to these, will ultimately be 
come both similar and equal among themselves. Q..E.D. 

COR. And hence in all reasonings about ultimate ratios, we may indif 
ferently use any one of those triangles for any other. 

LEMMA IX. 

If a ngnt line AE. and a curve tine ABC, both given by position, cut 
each other in a given angle, A ; and to that right line, in another 
given angle, BD, CE are ordinately applied, meeting the curve in B, 
C : and the points B and C together approach towards and meet in 
the point A : / say, that the areas of the triangles ABD, ACE, wilt 
ultimately be one to the other in the duplicate ratio of the sides. 




BOOK LI 



OF NATURAL PHILOSOPHY. 




For while the points B, C, approach 
towards the point A, suppose always AD 
to be produced to the remote points d and . 
e, so as Ad, Ae may be proportional to 
AD, AE ; and the ordinates db, ec, to be 
drawn parallel to the ordinates DB and 
EC, and meeting AB and AC produced D 
in b and c. Let the curve A be be similar 
to the curve A BC, and draw the right line 
Ag- so as to touch both curves in A, and 
cut the ordinates DB, EC, db ec, in F, G, 
J] g. Then, supposing the length Ae to remain the same, let the points B 
and C meet in the point A ; and the angle cAg vanishing, the curvilinear 
areas AW, Ace will coincide with the rectilinear areas A/rf, Age ; and 
therefore (by Lem. V) will be one to the other in the duplicate ratio of 
the sides Ad, Ae. But the areas ABD, ACE are always proportional to 
these areas ; and so the sides AD, AE are to these sides. And therefore 
the areas ABD, ACE are ultimately one to the other in the duplicate ratio 
of the sides AD, AE. Q.E.D. 

LEMMA X. 

The spaces which a bodij describes by any finite force urging it. whether 

that force is determined and immutable, or is continually augmented 

or continually diminished, are in the very beginning of the motion one 

to the other in the duplicate ratio of the times. 

Let the times be represented by the lines AD, AE, and the velocities 
generated in those times by the ordinates DB, EC. The spaces described 
with these velocities will be as the areas ABD, ACE. described by those 
ordinates, that is, at the very beginning of the motion (by Lem. IX), in 
the duplicate ratio of the times AD, AE. Q..E.D. 

COR. 1. And hence one may easily infer, that the errors of bodies des 
cribing similar parts of similar figures in proportional times, are nearly 
as the squares of the times in which they are generated ; if so be these 
errors are generated by any equal forces similarly applied to the bodies, 
and measured by the distances of the bodies from those places of the sim 
ilar figures, at which, without the action of those forces, the bodies would 
have arrived in those proportional times. 

COR. 2. But the errors that are generated by proportional forces, sim 
ilarly applied to the bodies at similar parts of the similar figures, are as 
the forces and the squares of the times conjuiu tly. 

COR. 3. The same thing is to be understood of any spaces whatsoever 
described by bodies urged with different forces ; all which, in the very be- 
g nning of the motion, are as the forces and the squares of the times conjunctly. 



100 



THE MATHEMATICAL PRINCIPLES 



I SEC. 1 



COR. 4. And therefore the forces are as the spaces described in the very 
beginning of the motion directly, and the squares of the times inversely. 

COR. 5. And the squares of the times are as the spaces described direct 
ly, und the forces inversely. 

SCHOLIUM. 

If in comparing indetermined quantities of different sorts one with 
another, any one is said to be as any other directly or inversely, the mean 
ing is, that the former is augmented or diminished in the same ratio with 
the latter, or with its reciprocal. And if any one is said to be as any other 
two or more directly or inversely, the meaning is, that the first is aug 
mented or diminished in the ratio compounded of the ratios in which the 
others, or the reciprocals of the others, are augmented or diminished. As 
if A is said to be as B directly, and C directly, and D inversely, the mean 
ing is, that A is augmented or diminished in the same ratio with B X C 
X -jj-, that is to say, that A and - arc one to the other in a given ratio. 

LEMMA XL 

The evanescent subtense of the angle of contact, in all curves which at 
the point of contact have a finite curvature, is ultimately in the dupli 
cate rati 1 ) of the subtense of the conterminate arc. 
CASE 1. Let AB be that arc, AD its tangent, BD 
the subtense of the angle of contact perpendicular on 
the tangent, AB the subtense of the arc. Draw BG 
perpendicular to the subtense AB, and AG to the tan 
gent AD, meeting in G ; then let the points D, B, and 
G. approach to the points d, b, and g, and suppose J 
to be the ultimate intersection of the lines BG, AG, 
when the points D, B, have come to A. It is evident 
that the distance GJ may be less than any assignable. 
But (from the nature of the circles passing through 
the points A, B, G, A, b, g,) AE 2 = AG X BD, and 
A6 2 = Ag X bd ; and therefore the ratio of AB 2 to Ab 2 is compounded oi 
the ratios of AG to Ag, and of Ed to bd. But because GJ may be as 
sumed of less length than any assignable, the ratio of AG to Ag may be 
such as to differ from the ratio of equality by less than any assignable 
difference ; and therefore the ratio of AB 2 to Ab 2 may be such as to differ 
from the ratio of BD to bd by less than any assignable difference. There 
fore, by Lem. I, the ultimate ratio of AB 2 to Ab 2 is the same with tho ul 
timate ratio of BD to bd. Q.E.D. 

CASE 2. Now let BD be inclined to AD in any given an*r1 r , and the 
ultimate ratio of BD to bd will always be the same as before, and there 
fore the same with the ratio of AB 2 to Ab 2 . Q.E-P 




BOOK I.] 



OF NATURAL PHILOSOPHY. 



101 



CASE 3. And if we suppose the angle D not to be given, but that the 
right line BD converges to a given point, or is determined by any other 
condition whatever ; nevertheless the angles D, d, being determined by the 
same law, will always draw nearer to equality, arid approach nearer to 
each other than by any assigned difference, and therefore, by Lem. I, will at 
lust be* equal ; and therefore the lines BD ; bd arc in the same ratio to each 
other as before. Q.E.D. 

COR. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, and 
their sines, BC, be, become ultimately equal to the chords AB, Ab } their 
squares will ultimately become as the subtenses BD, bd. 

COR. 2. Their squares are also ultimately as the versed sines of the arcs, 
bisecting the chords, and converging to a given point. For those versed 
sines are as the subtenses BD, bd. 

COR. 3. And therefore the versed sine is in the duplicate ratio of the 
time in which a body will describe the arc with a given velocity. 

COR. 4. The rectilinear triangles ADB, Adb are 
ultimately in the triplicate ratio of the sides AD, Ad, c 
and in a sesquiplicate ratio of the sides DB, db ; as 
being in the ratio compounded of the sides AD to DB, 
and of Ad to db. So also the triangles ABC, Abe 
are ultimately in the triplicate ratio of the sides BC, be. 
What I call the sesquiplicate ratio is the subduplicate 
of the triplicate, as being compounded of the simple 
and subduplicate ratio. j 

COR. 5. And because DB, db are ultimately paral- g 
lei and in the duplicate ratio of the lines AD, Ad, the 
ultimate curvilinear areas ADB, Adb will be (by the nature of the para 
bola) two thirds of the rectilinear triangles ADB, Adb and the segments 
AB, Ab will be one third of the same triangles. And thence those areas 
and those segments will be in the triplicite ratio as well of the tangents 
AD, Ad, as of the chords and arcs AB, AB. 

SCHOLIUM. 

But we have all along supposed the angle of contact to be neither infi 
nitely greater nor infinitely less than the angles of contact made by cir 
cles and their tangents ; that is, that the curvature at the point A is neither 
infinitely small nor infinitely great, or that the interval AJ is of a finite mag 
nitude. For DB may be taken as AD 3 : in which case no circle can be drawn 
through the point A, between the tangent AD and the curve AB, and 
therefore the angle of contact will be infinitely less than those of circles. 
And by a like reasoning, if DB be made successfully as AD 4 , AD 5 , AD 8 , 
AD 7 , etc., we shall have a series of angles of contact, proceeding in itifini- 
tum, wherein every succeeding term is infinitely less than the pre- 




102 THE MATHEMATICAL PRINCIPLES [BOOK 1 

ceding. And if DB be made successively as AD 2 , AD|, AD^, AD], AD| 
AD 7 , &c., we shall have another infinite series of angles of contact, the first 
of which is of the same sort with those of circles, the second infinitely 
greater, and every succeeding one infinitely greater than the preceding. 
But between any two of these angles another series of intermediate angles 
of contact may be interposed, proceeding both ways in infinitum. wherein 
every succeeding angle shall be infinitely greater or infinitely less than the 
preceding. As if between the terms AD 2 and AD 3 there were interposed 
the series AD f, ADy, AD 4 9 , AD|, AD?, AD|, AD^ 1 , AD^, AD^ 7 , &c. And 
again, between any two angles of this series, a new series of intermediate 
angles may be interposed, differing from one another by infinite intervals. 
Nor is nature confined to any bounds. 

Those things which have been demonstrated of curve lines, and the 
euperfices which they comprehend, may be easily applied to the curve su- 
perfices and contents of solids. These Lemmas are premised to avoid the 
tediousness of deducing perplexed demonstrations ad absurdnm, according 
to the method of the ancient geometers. For demonstrations are more 
contracted by the method of indivisibles : but because the hypothesis of 
indivisibles seems somewhat harsh, and therefore that method is reckoned 
less geometrical, I chose rather to reduce the demonstrations of the follow 
ing propositions to the first and last sums and ratios of nascent and evane 
scent quantities, that is, to the limits of those sums and ratios ; and so to 
premise, as short as I could, the demonstrations of those limits. For hereby 
the same thing is performed as by the method of indivisibles ; and now 
those principles being demonstrated, we may use them with more safety. 
Therefore if hereafter I should happen to consider quantities as made up of 
particles, or should use little curve lines for right ones, I would not be un- 
(lerstood to mean indivisibles, but evanescent divisible quantities : not the 
sums and ratios of determinate parts, but always the limits of sums and 
ratios ; and that the force of such demonstrations always depends on the 
method laid down in the foregoing Lemmas. 

Perhaps it may be objected, that there is no ultimate proportion, of 
evanescent quantities ; because the proportion, before the quantities have 
vanished, is not the ultimate, and when they are vanished, is none. But 
by the same argument, it may be alledged, that a body arriving at a cer 
tain place, and there stopping has no ultimate velocity : because the velo 
city, before the body comes to the place, is not its ultimate velocity ; when 
it has arrived, is none i ut the answer is easy; for by the ultimate ve 
locity is meant that with which the body is moved, neither before it arrives 
at its last place and the motion ceases, nor after, but at the very instant it 
arrives ; that is, that velocity with which the body arrives at its last place, 
and with which the motion ceases. And in like manner, by the ultimate ra 
tio of evanescent quantities is to Le understood the ratio of the ijuantitiea 



SEC. II.] OF NATURAL PHILOSOPHY. 103 

not before they vanish, nor afterwards, but with which they vanish. In 
like manner the first ratio of nascent quantities is that with which they begin 
to be. And the first or last sum is that with which they begin and cease 
to be (or to be augmented or diminished). There is a limit which the ve 
locity at the end of the motion may attain, but not exceed. This is the 
ultimate velocity. And there is the like limit in all quantities and pro 
portions that begin and cease to be. And since such limits are certain and 
definite, to determine the same is a problem strictly geometrical. But 
whatever is geometrical we may be allowed to use in determining and de 
monstrating any other thing that is likewise geometrical. 

It may also be objected, that if the ultimate ratios of evanescent quan 
tities are given, their ultimate magnitudes will be also given : and so all 
quantities will consist of indivisibles, which is contrary to what Euclid 
has demonstrated concerning incommensurables, in the 10th Book of his 
Elements. But this objection is founded on a false supposition. For 
those ultimate ratios with which quantities vanish are not truly the ratios 
of ultimate quantities, but limits towards which the ratios of quantities 
decreasing without limit do always converge ; and to which they approach 
nearer than by any given difference, but never go beyond, nor in effect attain 
to, till the quantities are diminished in wfinitum. This thing will appear 
more evident in quantities infinitely great. If two quantities, whose dif 
ference is given, be augmented in infin&um, the ultimate ratio of these 
quantities will be given, to wit, the ratio of equality ; but it does not from 
thence follow, that the ultimate or greatest quantities themselves, whose 
ratio that is, will be given. Therefore if in what follows, for the sake of 
being more easily understood, I should happen to mention quantities as 
least, or evanescent, or ultimate, you are not to suppose that quantities of 
any determinate magnitude are meant, but such as are conceived to be al 
ways diminished without end. 



SECTION II. 

Of the Invention of Centripetal Forces. 

PROPOSITION I. THEOREM 1. 

The areas, which revolving bodies describe by radii drawn to an ^mmo- 

vable centra of force do lie in tJ:e same immovable planes, and are pro- 

portional to the times in which they are described. 

For suppose the time to be divided into equal parts, and in the first part 

of that time let the body by its innate force describe the right line AB 

In the second part of that time, the same would (by Law I.), if not hindered, 

proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radii 

AS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de- 



104 



THE MATHEMATICAL PRINCIPLES 



[BOOK I 




scribed. But when the body 
is arrived at B, suppose 
that a centripetal force acts 
at once with a great im 
pulse, and, turning aside the 
body from the right line Be, 
compels it afterwards to con 
tinue its motion along the 
right line BC. Draw cC 
parallel to BS meeting BC 
in C ; and at the end of the 
second part of the time, the 
body (by Cor. I. of the Laws) 
will be found in C, in the 
same plane with the triangle 
A SB. Join SC, and, because s 
SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc, 
and therefore also to the triangle SAB. By the like argument, if the 
centripetal force acts successively in C, D, E. &c., and makes the body, in 
each single particle of time, to describe the right lines CD, DE, EF 7 &c., 
they will all lie in the same plane : and the triangle SCD will be equal to 
the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, 
in equal times, equal areas are described in one immovable plane : and, by 
composition, any sums SADS, SAFS, of those areas, are one to the other 
as the times in which they are described. Now let the number of those 
triangles be augmented, and their breadth diminished in wjinitum ; and 
(by Cor. 4, Lem. III.) their ultimate perimeter ADF will be a curve line : 
and therefore the centripetal force, by which the body is perpetually drawn 
back from the tangent of this curve, will act continually ; and any described 
areas SADS, SAFS, which are always proportional to the times of de 
scription, will, in this case also, be proportional to those times. Q.E.D. 

COR. 1. The velocity of a body attracted towards an immovable centre, 
in spaces void of resistance, is reciprocally as the perpendicular let fall 
from that centre on the right line that touches the orbit. For the veloci 
ties in those places A, B, C, D, E. are as the bases AB, BC, CD, DE, EF. 
of equal triangles ; and these bases are reciprocally as the perpendiculars 
let fall upon them. 

COR. 2. If the chords AB, BC of two arcs, successively described in 
equal times by the same body, in spaces void of resistance, are completed 
into a parallelogram ABCV, and the diagonal BV of this parallelogram; 
in the position which it ultimately acquires when those arcs are diminished 
in irifinitum, is produced both ways, it will pass through the centre of force. 

COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equal 



SEC. II.] 



OF NATURAL PHILOSOPHY. 



105 



times, in spaces void of resistance, are completed into the parallelograms 
ABCV, DEFZ : the forces in B and E are one to the other in the ulti 
mate ratio of the diagonals BV, EZ, when those arcs are diminished in 
infinitum. For the motions BC and EF of the body (by Cor. 1 of the 
Laws) are compounded of the motions Be, BV, and E/", EZ : but BV and 
EZ, which are equal to Cc and F/, in the demonstration of this Proposi 
tion, were generated by the impulses of the centripetal force in B and E ; 
and are therefore proportional to those impulses. 

COR. 4. The forces by which bodies, in spaces void of resistance, are 
drawn back from rectilinear motions, and turned into curvilinear orbits, 
are one to another as the versed sines of arcs described in equal times ; which 
versed sines tend to the centre of force, and bisect the chords when those 
arcs are diminished to infinity. For such versed sines are the halves of 
the diagonals mentioned in Cor. 3. 

COR. 5. And therefore those forces are to the force of gravity as the said 
versed sines to the versed sines perpendicular to the horizon of those para 
bolic arcs which projectiles describe in the same time. 

COR. 6. And the same things do all hold good (by Cor. 5 of the Laws), 
when the planes in which the bodies are moved, together with the centres 
of force which are placed in those planes, are not at rest, but move uni 
formly forward in right lines. 

PROPOSITION II. THEOREM II. 

Every body that moves in any curve line described in a plane, and by a 
radius, drawn to a point either immovable, or moving forward with 
an uniform rectilinear motion, describes about that point areas propor 
tional to the times, is urged by a centripetal force directed to that point 
CASE. 1. For every body 

that moves in a curve line, 

is (by Law 1) turned aside 

from its rectilinear course 

by the action of some force 

that impels it. And that force 

by which the body is turned 

off from its rectilinear course, 

and is made to describe, in 

equal times, the equal least 

triangles SAB, SBC, SCD, 

&c., about the immovable 

point S (by Prop. XL. Book 

1, Elem. and Law II), acts 

in the place B, according to 



the direction of a line 



par- 




1U6 THE MATHEMATICAL PRINCIPLES [BOOK f. 

allel K cC. that is, in the direction of the line BS. and in the place C, 
accordii g to the direction of a line parallel to dD, that is, in the direction 
of the line CS, (fee.; and therefore acts always in the direction of lines 
tending to the immovable point S. Q.E.I). 

CASE. 2. And (by Cor. 5 of the Laws) it is indifferent whether the su- 
perfices in which a body describes a curvilinear figure be quiescent, or moves 
together with the body, the figure described, and its point S, uniformly 
forward in right lines. 

COR. 1. In non-resisting spaces or mediums, if the areas are not propor 
tional to the times, the forces are not directed to the point in which the 
radii meet ; but deviate therefrom in. consequently or towards the parts to 
which the motion is directed, if the description of the areas is accelerated ; 
but in antecedentia, if retarded. 

COR. 2. And even in resisting mediums, if the description of the areas 
is accelerated, the directions of the forces deviate from the point in which 
the radii meet, towards the parts to which the motion tends. 

SCHOLIUM. 

A body may be urged by a centripetal force compounded of several 
forces ; in which case the meaning of the Proposition is, that the force 
which results out of all tends to the point S. But if any force acts per 
petually in the direction of lines perpendicular to the described surface, 
this force will make the body to deviate from the plane of its motion : but 
will neither augment nor diminish the quantity of the described surface 
and is therefore to be neglected in the composition of forces. 

PROPOSITION III. THEOREM III. 

Every body, that by a radius drawn to the centre of another body, how 
soever moved, describes areas about that centre proportional to iJie times, 
is urged by a force compounded out of the centripetal force Bending fo 
that other body, and of all the accelerative force by which that other 
body is impelled. 

Let L represent the one, and T the other body ; and (by Cor. of the Laws) 
if both bodies are urged in the direction of parallel lines, by a ne T force 
equal and contrary to that by which the second body T is tinned, the first 
body L will go on to describe about the other body T the same areas as 
before : but the force by which that other body T was urged will be now 
destroyed by an equal and contrary force; and therefore (by Law I.) that 
other body T, now left to itself, will either rest, or move uniformly forward 
in a right line : and the first body L impelled by the difference of the 
forces, that is, by the force remaining, will go on to describe about the other 
body T areas proportional to the times. And therefore (by Theor. II.) the 
difference ;f the forces is directed to the other body T as its centre. Q.E.D 



SEC. IL] OF NATURAL PHILOSOPHY. 107 

Co.*. 1. Hence if the one body L, by a radius drawn to the other body T, 
describes areas proportional to the times ; and from the whole force, by which 
the firr.t body L is urged (whether that force is simple, or, according to 
Cor. 2 of the Laws, compounded out of several forces), we subduct (by the 
same Cor.) that whole accelerative force by which the other body is urged ; 
the who_e remaining force by which the first body is urged will tend to the 
( ther body T, as its centre. 

COR. 2. And, if these areas are proportional to the times nearly, the re 
maining force will tend to the other body T nearly. 

COR. 3. And vice versa, if the remaining force tends nearly to the other 
body T, those areas will be nearly proportional to the times. 

COR. 4. If the body L, by a radius drawn to the other body T, describes 
areas, which, compared with the times, are very unequal ; and that other 
body T be either at rest, or moves uniformly forward in a right line : the 
action of the centripetal force tending to that other body T is either none 
at all, or it is mixed and compounded with very powerful actions of other 
forces : and the whole force compounded of them all, if they are many, is 
directed to another (immovable or moveaJble) centre. The same thing ob 
tains, when the other body is moved by any motion whatsoever ; provided 
that centripetal force is taken, w r hich remains after subducting that whole 
force acting upon that other body T. 

SCHOLIUM. 

Because the equable description of areas indicates that a centre is re 
spected by that force with which the body is most affected, and by which it 
is drawn back from its rectilinear motion, and retained in its orbit ; why 
may we not be allowed, in the following discourse, to use the equable de 
scription of areas as an indication of a centre, about which all circular 
motion is performed in free spaces ? 

PROPOSITION IV. THEOREM IV. 

The centripetal forces of bodies, which by equable motions describe differ 
ent circles, tend to the centres of the same circles ; and are one to tJie 
other as the squares of t/ie arcs described in equal times applied to the 
radii of the circles. 

These forces tend to the centres of the circles (by Prop. II., and Cor. 2, 
Prop. L), and are one to another as the versed sines of the least arcs de 
scribed in equal times (by Cor. 4, Prop. I.) ; that is, as the squares of the 
same arcs applied to the diameters of the circles (by Lem. VII.) ; and there 
fore since those arcs are as arcs described in any equal times, and the dia- 
me ers ace as the radii, the forces will be as the squares of any arcs de- 
scr bed in the same time applied to the radii of the circles. Q.E.D. 
^OR. 1. Therefore, since those arcs are as the velocities of the bodies. 



I OS THE MATHEMATICAL PRINCIPLES [BOOK . 

the centripetal forces are in a ratio compounded of the duplicate ra jio of 
the velocities directly, and of the simple ratio of the radii inversely. 

COR. 2. And since the periodic times are in a ratio compounded of the 
ratio of the radii directly, and the ratio of the velocities inversely, the cen 
tripetal forces, are in a ratio compounded of the ratio of the radii directly, 
and the duplicate ratio of the periodic times inversely. 

COR, 3. Whence if the periodic times are equal, and the velocities 
therefore as the radii, the centripetal forces will be also as the radii ; and 
tke contrary. 

COR. 4. If the periodic times and the velocities are both in the subdu- 
plicate ratio of the radii, the centripetal forces will be equal among them 
selves ; and the contrary. 

COR. 5. If the periodic times are as the radii, and therefore the veloci 
ties equal, the centripetal forces will be reciprocally as the radii ; and the 
contrary. 

COR. 6. If the periodic times are in the sesquiplicate ratio of the radii, 
and therefore the velocities reciprocally in the subduplicate ratio of the 
radii, the centripetal forces will be in the duplicate ratio of the radii in 
versely : and the contrary. 

COR. 7. And universally, if the periodic time is as any power R n of the 
radius R, and therefore the velocity reciprocally as the power R n ] of 
the radius, the centripetal force will be reciprocally as the power R 2n 1 of 
the radius; and the contrary. 

COR. 8. The same things all hold concerning the times, the velocities, 
and forces by which bodies describe the similar parts of any similar figures 
that have their centres in a similar position with those figures ; as appears 
by applying the demonstration of the preceding cases to those. And the 
application is easy, by only substituting the equable description of areas in 
the place of equable motion, and using the distances of the bodies from the 
centres instead of the radii. 

COR. 9. From the same demonstration it likewise follows, that the arc 
which a body, uniformly revolving in a circle by means of a given centri 
petal force, describes in any time, is a mean proportional between the 
diameter of the circle, and the space which the same body falling by the 
same given force would descend through in the same given time. 

SCHOLIUM. 

The case of the 6th Corollary obtains in the celestial bodies (as Sir 
Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed) ; 
and therefore in what follows, I intend to treat more at large of those 
things which relate to centripetal force decreasing in a duplicate ratio 
of the distances from the centres. 

Moreover, by means of the preceding Proposition and its Corollaries, we 



SEC. II.] OF NATURAL PHILOSOPHY. 109 

may discover the proportion of a centripetal force to any other known 
force, such as that of gravity. For if a body by means of its gravity re 
volves in a circle concentric to the earth, this gravity is the centripetal 
force of that body. But from the descent of heavy bodies, the time of one 
entire revolution, as well as the arc described in any given time, is given 
(by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his 
excellent book De Horologio Oscillatorio, has compared the force of 
gravity with the centrifugal forces of revolving bodies. 

The preceding Proposition may be likewise demonstrated after this 
manner. In any circle suppose a polygon to be inscribed of any number 
of sides. And if a body, moved with a given velocity along the sides of the 
polygon, is reflected from the circle at the several angular points, the force, 
with which at every reflection it strikes the circle, will be as its velocity : 
and therefore the sum of the forces, in a given time, will be as that ve 
locity and the number of reflections conjunctly ; that is (if the species of 
the polygon be given), as the length described in that given time, and in 
creased or diminished in the ratio of the same length to the radius of the 
circle ; that is, as the square of that length applied to the radius ; and 
therefore the polygon, by having its sides diminished in inftnitum, coin 
cides with the circle, as the square of the arc described in a given time ap 
plied to the radius. This is the centrifugal force, with which the body 
impels the circle ; and to which the contrary force, wherewith the circle 
continually repels the body towards the centre, is equal. 

PROPOSITION V. PROBLEM I. 

There being given, in any places, the velocity with which a body de 
scribes a given figure, by means of forces directed to some common 
centre : to find that centre. 

Let the three right lines PT, TQV, VR 
touch the figure described in as many points, 
P, Q, R, and meet in T and V. On the tan 
gents erect the perpendiculars PA, QB, RC, 
reciprocally proportional to the velocities of the 
body in the points P, Q, R, from which the 
perpendiculars were raised ; that is, so that PA 
may be to QB as the velocity in Q to the velocity in P, and QB to RC 
as the velocity in R to the velocity in Q. Through the ends A, B, C, of 
the perpendiculars draw AD, DBE, EC, at right angles, meeting in D and 
E : and the right lines TD, VE produced, will meet in S, the centre re 
quired. 

For the perpendiculars let fall from the centre S on the tangents PT. 
QT. are reciprocally as the velocities of the bodies in the points P and Q 




110 THE MATHEMATICAL PRINCIPLES [BOOK 1 

(by Cor. 1, Prop. I.), and therefore, by construction, as the perpendiculars 
AP, BQ, directly ; that is, as the perpendiculars let fall from the point D 
on the tangents. Whence it is easy to infer that the points S, D, T, are 
in one right line. And by the like argument the points S, E, V are also 
in one right line ; and therefore the centre S is in the point where the 
right lines TD ; YE meet. Q.E.D. 

PROPOSITION VL THEOREM V. 

In a space void of resistance, if a body revolves in any orbit about an im 
movable centre, and in the least time describes any arc just then, na 
scent ; and the versed sine of that arc is supposed to be drawn bisect 
ing the chord, and produced passing through the centre of force: the 
centripetal force in the middle of the arc will be as the versed sine di 
rectly and the square of the time inversely. 
For the versed sine in a given time is as the force (by Cor. 4, Prop. 1) ; 

and augmenting the time in any ratio, because the arc will be augmented 

in the same ratio, the versed sine will be augmented in the duplicate of 

that ratio (by Cor. 2 and 3, Lem. XL), and therefore is as the force and the 

square of the time. Subduct on both sides the duplicate ratio of the 

time, and the force will be as the versed sine directly, arid the square of 

the time inversely. Q.E.D. 

And the same thing may also be easily demonstrated by Corol. 4 ? 

T,em. X. 

COR. 1. If a body P revolving about the 

centre S describes a curve line APQ,, which a 

right line ZPR touches in any point P ; and 

from any other point Q, of the curve, QJl is 

drawn parallel to the distance SP, meeting 

the tangent in R ; and QT is drawn perpen- 

(licular to the distance SP ; the centripetal force will be reciprocally as the 

sp 2 x Q/r 2 

solid - : , if the solid be taken of that magnitude which it ulti- 




mately acquires when the points P and Q, coincide. For Q,R is equal to 
the versed sine of double the arc QP, whose middle is P : and double the 
triangle SQP, or SP X Q,T is proportional to the time in which that 
double arc is described ; and therefore may be used for the exponent of 
the time. 

COR. 2. By a like reasoning, the centripetal force is reciprocally as the 

SY 2 X QJP 2 
solid - 7^5 - ; if SY is a perpendicular from the centre of force on 



PR the tangent of the orbit. For the rectangles SY X QP and SP X Q,T 
are equal. 



SEC. II.] OF NATURAL PHILOSOPHY. Ill 

COR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncen- 
trically, that is, contains with a circle the least angle of contact or sec 
tion, having the same curvature rnd the same radius of curvature at the 
point P : and if PV be a chord of this circle, drawn from the body through 
the centre of force ; the centripetal force will be reciprocally as the solid 

QP 2 

SY 2 X PV. For PV is - . 



COR. 4. The same things being supposed, the centripetal force is as the 
square of the velocity directly, and that chord inversely. For the velocity 
is reciprocally as the perpendicular SY, by Cor. 1. Prop. I. 

COR. 5. Hence if any curvilinear figure APQ, is given, and therein a 
point S is also given, to which a centripetal force is perpetually directed. 
that law of centripetal force may be found, by which the body P will bc j 
continually drawn back from a rectilinear course, and. being detained in 
the perimeter of that figure, will describe the same by a perpetual revolu- 

SP 2 x QT 2 

tion. That is, we are to find, by computation, either the solid ---- - 



or the solid SY 2 X PV, reciprocally proportional to this force. Example: 
of this we shall give in the following Problems. 

PROPOSITION VII. PROBLEM II. 

Tf a body revolves in the circumference of a circle; it is proposed to finii 
the law of centripetal force directed to any given, point. 

Let VQPA be the circumference of the 
circle ; S the given point to which as to 
a centre the force tends : P the body mov 
ing in the circumference ; Q the next 
place into which it is to move; and PRZ 
the tangent of the circle at the preceding 
place. Through the point S draw the v 
chord PV, and the diameter VA of the 
circle : join AP, and draw Q,T perpen 
dicular to SP, which produced, may meet 
the tangent PR in Z ; and lastly, through 
the point Q, draw LR parallel to SP, meeting the circle" in L, and the 
tangent PZ in R. And, because of the similar triangles ZQR, ZTP. 
VPA, we shall have RP 2 , that is. QRL to QT 2 as AV 2 to PV 2 . And 

QRlj x PV 2 SI 3 - 

therefore - TS -- is equal to QT 2 . Multiply those equals by - . 




and the points P and Q, coinciding, for RL write PV ; then we shall have 

SP- X PV 5 SP 2 x QT 2 

And therefore flr Cor 1 and 5. Prop. VI.) 



112 THE MATHEMATICAL PRINCIPLES [BOOK I, 

SP 2 X PV 3 

the centripetal force is reciprocally as - ry^~ J that is (because AV 2 



ia given), reciprocally as the square of the distance or altitude SP, and the 
3ube of the chord PV conjunctly. Q.E.L 

The same otherwise. 

On the tangent PR produced let fall the perpendicular SY ; and (be 
cause of the similar triangles SYP, VPA), we shall have AV to PV as SP 

SP X PV SP 2 >< PV 3 

to SY, and therefore -- ^~ - = SY, and - ^- = SY 2 X PV. 
A V A V 

And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip- 

SP 2 X PV 3 

rocally as - ~~ry~~~ I * na * * s (because AV is given), reciprocally as SP" 

X PV 3 . Q.E.I. 

Con. 1. Hence if the given point S, to which the centripetal force al 
ways tends, is placed in the circumference of the circle, as at V, the cen 
tripetal force will be reciprocally as the quadrato-cube (or fifth power) of 
the altitude SP. 

COR. 2. The force by which the body P in the 
circle APTV revolves about the centre of force S 
is to the force by which the same body P may re 
volve in the same circle, and in the same periodic 
time, about any other centre of force R, as RP 2 X 
SP to the cube of the right line SG, which, from 
the first centre of force S is drawn parallel to the 
distance PR of the body from the second centre of force R, meeting the 
tangent PG of the orbit in G. For by the construction of this Proposition, 
the former force is to the latter as RP 2 X PT 3 to SP 2 X PV 3 ; that is, as 

SP 3 X PV 3 

SP X RP 2 to -- p ; or (because of the similar triangles PSG, TPV) 

to SG S . 

COR. 3. The force by which the body P in any orbit revolves about the 
centre of force S, is to the force by which the same body may revolve in 
the same orbit, and the same periodic time, about any other centre of force 
R. as the solid SP X RP 2 , contained under the distance of the body from 
the first centre of force S, and the square of its distance from the sec 
ond centre of force R, to the cube of the right line SG, drawn from the 
first centre of the force S, parallel to the distance RP of the body from 
fch*3 second centre of force R, meeting the tangent PG of the orbit in G. 
For the force in this orbit at any point P is the same as in a circle of the 
same curvature. 




SJSG. IL] 



OF NATURAL PHILOSOPHY. 



113 



PROPOSITION VIII. PROBLEM III. 

If a body mi ues in the semi-circuwferencePQA: it is proposed to find 
the law of the centripetal force tending to a point S, so remote, that all 
the lines PS. RS drawn thereto, may be taken for parallels. 
From C, the centre of the semi-circle, let 

the semi-diameter CA he drawn, cutting the 

parallels at right angles in M and N, and 

join CP. Because of the similar triangles 

CPM, PZT, and RZQ, we shall have CP 2 

to PM 2 as PR 2 to QT 2 ; and, from the na 

ture of the circle, PR 2 is equal to the rect 

angle QR X RN + QN, or, the points P, Q coinciding, to the rectangle 

QR x 2PM. Therefore CP 2 is to PM 2 as QR X 2PM to QT 2 ; and 

QT 2 2PM 3 QT 2 X SP 2 2PM 3 X SP 2 

therefore (by 




QR 

Corol. 
8PM 3 X SP 2 



, 
and 



QR 



And 



1 and 5, Prop. VI.), the centripetal force is reciprocally as 

2SP 2 



. 
that is (neglecting the given ratio -ppr)> reciprocally as 

PM 3 . Q.E.L 

And the same thing is likewise easily inferred from the preceding Pro 
position. 

SCHOLIUM. 

And by a like reasoning, a body will be moved in an ellipsis, or even ia 
an hyperbola, or parabola, by a centripetal force which is reciprocally ae 
the cube of the ordinate directed to an infinitely remote centre of force. 

PROPOSITION IX. PROBLEM IV. 

If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc., 
in a given angle; it is proposed to find thelaio of the centripetal force 
tending to tJie centre of that spiral. 
Suppose the inde 

finitely small angle AY 

PSQ to be given ; be 

cause, then, all the 

angles are given, the 

figure SPRQT will , _ 

be given in specie. v 

QT Q,T 2 

Therefore the ratio -7^- is also given, and is as QT, that is (be 
lot IX QK 

cause the figure is given in specie), as SP. But if the angle PSQ is any 
way changed, the right line QR, subtending the angle of contact QPU 




tU 



THE MATHEMATICAL PRINCIPLES 



[BOOK J 



(by Lemma XI) will be changed in the duplicate ratio of PR or QT 

QT 2 

Therefore the ratio ~TVD~ remains the same as before, that is, as SP. And 

QT 2 x SP 2 

-^ is as SP 3 , and therefore (by Corol. 1 and 5, Prop. YI) the 

centripetal force is reciprocally as the cube of the distance SP. Q.E.I. 

The same otherwise. 

The perpendicular SY let fall upon the tangent, and the chord PY of 
the circle concentrically cutting the spiral, are in given ratios to the height 
SP ; and therefore SP 3 is as SY 2 X PY, that is (by Corol. 3 and 5, Prop. 
YI) reciprocally as the centripetal force. 

LEMMA XII. 

All parallelograms circumscribed about any conjugate diameters of a 
given ellipsis or hyperbola are equal among themselves. 
This is demonstrated by the writers on the conic sections. 

PROPOSITION X. PROBLEM Y. 

If a body revolves in an ellipsis ; it is proposed to find the law of thi 
centripetal force tending to the centre of the ellipsis. 

Suppose CA, CB to 
be semi-axes of the 
ellipsis; GP, DK, con 
jugate diameters ; PF, 
Q,T perpendiculars to 
those diameters; Qvan 
^rdinate to the diame 
ter GP ; and if the 
parallelogram QvPR 
be completed, then (by 
the properties of the 
jonic sections) the rec- 
langle PvG will be to 
Qv 2 as PC 2 to CD 2 ; 
and (because of the 
similar triangles Q^T, PCF), Qi> 2 to QT 2 as PC 2 to PF 2 ; and, by com 
position, the ratio of PvG to QT 2 is compounded of the ratio of PC 2 1< 

QT 2 
CD 2 , and of the ratio of PC 2 to PF 2 , that is, vG to -p as PC ; 

to _92L^_ P _ ] ^_. Put QR for Pr, and (by Lem. XII) BC X CA for CD 

K PF ; also (the points P and Q coinciding) 2PC for rG; and multiply- 




SEC. II.] OF NATURAL PHILOSOPHY. 115 

QT 2 x PC 2 

ing the extremes and means together, we shall have rfo~ equal to 

2BC 2 X CA 2 

pp . Therefore (by Cor. 5, Prop. VI), the centripetal force is 

2BC 2 X CA 2 

reciprocally as ry~ ; that is (because 2I3C 2 X CA 2 is given), re 
ciprocally as-r^v; that is, directly as the distance PC. QEI. 
I O 

TJie same otherwise. 

[n the right line PG on the other side of the point T, take the point u 
so that Tu may be equal to TV ; then take uV, such as shall be to v G as 
DC 2 to PC 2 . And because Qr 9 is to PvG as DC 2 to PC 2 (by the conic 
sections), we shall have Qv 2 -= Pi X V. Add the rectangle n.Pv to both 
sides, and the square of the chord of the arc PQ, will be equal to the rect 
angle VPv ; and therefore a circle which touches the conic section in P, 
and passes through the point Q,, will pass also through the point V. Now 
let the points P and Q, meet, and the ratio of nV to rG, which is the same 
with the ratio of DC 2 to PC 2 , will become the ratio of PV to PG, or PV 

2DC 2 

to 2PC : and therefore PY will be equal to . And therefore the 

force by which the body P revolves in the ellipsis will be reciprocally as 

2 DC 2 

ry X PF 2 (by Cor. 3, Prop. VI) ; that is (because 2DC 2 X PF 2 is 
I O 

given) directly as PC. Q.E.I. 

COR. 1. And therefore the force is as the distance of the body from the 
centre of the ellipsis ; and, vice versa, if the force is as the distance, the 
body will move in an ellipsis whose centre coincides with the centre of force, 
or perhaps in a circle into which the ellipsis may degenerate. 

COR. 2. And the periodic times of the revolutions made in all ellipses 
whatsoever about the same centre will be equal. For those times in sim 
ilar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipses 
that have their greater axis common, they are one to another as the whole 
areas of the ellipses directly, and the parts of the areas described in the 
same time inversely: that is, as the lesser axes directly, and the velocities 
of the bodies in their principal vertices inversely ; :hat is, as those lesser 
axes dirtily, and the ordinates to the same point % f the common axes in 
versely ; and therefore (because of the equality of the direct and inverse 
ratios) in the ratio of equality. 

SCHOLIUM. 

If the ellipsis, by having its centre removed to an infinite distance, de 
generates into a parabola, the body will move in tin s parabola ; and the 



116 



THE MATHEMATICAL PRINCIPLES 



[BOOK I 



force, now tending to a centre infinitely remote, will become equable. 
Which is Galileo s theorem. And if the parabolic section of the cone (by 
changing the inclination of the cutting plane to the cone) degenerates into 
an hyperbola, the body will move in the perimeter of this hyperbola, hav 
ing its centripetal force changed into a centrifugal force. And in like 
manner as in the circle, or in the ellipsis, if the forces are directed to the 
centre of the figure placed in the abscissa, those forces by increasing or di 
minishing the ordinates in any given ratio, or even by changing the angle 
of the inclination of the ordinates to the abscissa, are always augmented 
or diminished in the ratio of the distances from the centre ; provided the 
periodic times remain equal ; so also in all figures whatsoever, if the ordi- 
nates are augmented or diminished in any given ratio, or their inclination 
is any way changed, the periodic time remaining the same, the forces di 
rected to any centre placed in the abscissa are in the several ordinatee 
augmented or diminished in the ratio of the distances from the centre 



SECTION III. 

Of the motion of bodies in eccentric conic sections. 

PROPOSITION XL PROBLEM VI. 

If a body revolves in an ellipsis ; it is required to find the law of the 

centripetal force tending to the focus of the ellipsis. 

Let S be the focus 
of the ellipsis. Draw 
SP cutting the diame 
ter DK of the ellipsis 
in E, and the ordinate 
Qv in x ; and com 
plete the parallelogram 
d.rPR, It is evident 
that EP is equal to the 
greater semi-axis AC : 
for drawing HI froln 
the other focus H of 
the ellipsis parallel to 
EC, because CS, CH 
are equal, ES, El will 
be also equal ; so that EP is the half sum of PS, PI, that is (because of 
the parallels HI, PR, and the equal angles IPR, HPZ), of PS, PH, which 
taken together are equal to the whole axis 2AC. Draw QT perpendicu 
lar to SP, and putting L for the princi al latus rectum of the ellipsis (or for 




III. OF NATURAL PHILOSOPHY. 117 



L X ^ R t0 L X Py aS ^ R t0 PV that 1S > US PE 



or AC to PC ; and L X Pv to GvP as L to Gy ; and GvP to Qi> 2 as 
to CD- ; and by (Corol. 2, Lem. VII) the points Q, and P coinciding, Qv* 
is to Q,r- in the ratio of equality ; and Q,.r 2 or Qv 2 is to Q,T 2 as EP 2 to 
PF 2 , that is, as CA 2 to PF 2 , or (by Lem. XII) as CD 2 to CB 2 . And com 
pounding all those ratios together, we shall have L X QR to Q,T 2 as AC 
X L X PC 2 X CD 2 , or 2CB 2 X PC 2 X CD 2 to PC X Gv X CD 2 X 
CB 2 , or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gr 
are equal. And therefore the quantities L X QR and Q,T 2 , proportional 

SP 2 

to these, will be also equal. Let those equals be drawn into-p^B"? and L 



SP 2 X QT 2 

X SP 2 will become equal to -- ^p . And therefore (by Corol. 1 and 



5, Prop. VI) the centripetal force is reciprocally as L X SP 2 , that is, re 
ciprocally in the duplicate ratio of the distance SP. Q.E.I. 

The same otherwise. 

Since the force tending to the centre of the ellipsis, by which the body 
P may revolve in that ellipsis, is (by Corol. 1, Prop. X.) as the distance 
CP of the body from the centre C of the ellipsis ; let CE be drawn paral 
lel to the tangent PR of the ellipsis : and the force by which the same body 
P may revolve about any other point S of the ellipsis, if CE and PS in- 

PE 3 

tersect in E, will be as ^T 3 , (by Cor. 3, Prop. VII.) ; that is, if the point 

S is the focus of the ellipsis, and therefore PE be given as SP 2 recipro 
cally. Q.E.I. 

With the same brevity with which we reduced the fifth Problem to the 
parabola, and hyperbola, we might do the like here : but because of the 
dignity of the Problem and its use in what follows, I shall confirm the other 
cases by particular demonstrations. 

PROPOSITION XII. PROBLEM VII. 

Suppose a body to move in an hyperbola ; it is required to find lite law of 

the centripetal force tending to the focus of that figure. 

Let CA, CB be the semi-axes of the hyperbola ; PG, KD other con 

jugate diameters ; PF a perpendicular to the diameter KD ; and Qv an 

ordinate to the diameter GP. Draw SP cutting the diameter DK in E, 

and the ordinate Qv in x, and complete the parallelogram QRP.r. It is 

evident that EP is equal to the semi-transverse axis AC ; for drawing 

HE, from the other focus H of the hyperbola, parallel to EC, because CS, 

TH are equal, ES El will be also equal ; so that EP is the half difference 



J1S 



THE MATHEMATICAL PRINCIPLES 



[Book I 



.of PS, PI; that is (be 
cause of the parallels IH, 
PR, and the equal angles 
IPR, HPZ), of PS, PH, 
the difference of which is 
equal to the whole axis 
2AC. Draw Q,T perpen 
dicular to SP; and put 
ting L for the principal 
latus rectum of the hy 
perbola (that is, for 

2BC 2 \ .... 

-Tp- ) 7 we shall have L 

X QR to L X Pv as QR 
to Pv, or Px to Pv, that is 
(because of the similar tri 
angles Pxv, PEC), as PE 
to PC, or AC to PC. 
And L X Pv will be to 
Gv X Pv as L to Gv; 
and (by the properties of 
the conic sections) the rec 
tangle G? P is to Q,v 2 as 
PC 2 to CD 2 ; and by (Cor. 2, Lem. VII.), Qv 2 to Qa* the points Q and P 
coinciding, becomes a ratio of equality ; and Q,.r 2 or Qv 2 is to Q,T 2 as EP 2 
to PF 2 , that is, as CA 2 to PF 2 , or (by Lem. XII.) as CD 2 to CB 2 : and, 
compounding all those ratios together, we shall have L X Q,R to Q,T 2 as 
AC X L X PC 2 X CD 2 , or 2CB 2 X PC 2 X CD 2 to PC X Gv X CD 2 
X CB 2 , or as 2PC to Gv. But the points P and Q, coinciding. 2PC and 
Gv are equal. And therefore the quantities L X Q,R arid Q.T 2 , propor 
tional to them, will be also equal. Let those equals be drawn into 

SP 2 sp 2 x o/r 2 

^, and we shall have L X SP 2 equal to ^^ . And therefore (by 




Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP 2 . 
hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I. 

TJie same otherwise. 

Find out the force tending from the centre C of the hype rbola. This will 
be proportional to the distance CP. But from thence (by Cor. 3, Prop. 

PE 3 
VII.) the force tending to the focus S will be as -^-^ th; (t is, because PE 

is given reciprocally as SP-. Q,.E.I. 



SEC. III.] 



OF NATURAL PHILOSOPHY. 



119 




And the same way may it be demonstrated, that the body having its cen 
tripetal changed into a centrifugal force, will move in the conjugate hy 
perbola. 

LEMMA XIII. 

The latus rectum of a parabola belonging to any vertex is quadruple 
the distance of that vertex from the focus of thejigurc. 

This is demonstrated by the writers on the conic sections. 

LEMMA XIV. 

Tlie perpendicular, let fall from the focus of a parabola on its tangent, is 
a mean proportional between the. distances of the focus from the poini 
of contact, and from the principal vertex of the figure. 

For, let AP be the parabola, S its 
focus, A its principal vertex, P the 
point of contact, PO an ordinate to the 
principal diameter. PM the tangent 
meeting the principal diameter in M. 

and SN the perpendicular from the fo-~ M A s o 

cus on the tangent : join AN, and because of the equal lines MS and SP, 
MN and NP, MA and AC, the right lines AN, OP, will be parallel ; and 
thence the triangle SAN will be right-angled at A, and similar to the 
equal triangles SNM, SNP j therefore PS is to SN as SN to SA. Q.E.D. 

COR. 1. PS 2 is to SN 2 as PS to SA. 

COR. 2. And because SA is given, SN- will be as PS. 

COR. 3. And the concourse of any tangent PM, with the right line SN. 
drawn from the focus per] endicular on the tangent, falls in the right line 
AN that touches the parabola in the principal vertex. 

PROPOSITION XIII. PROBLEM VIII. 

If a body moves in the perimeter of a parabola ; it is required to find the. 

law of the centripetal force tending to the focus of that figure. 

Retaining the construction 
of the preceding Lemma, let P 
be the body in the perimeter 
of the parabola ; and from the 
place Q,, into which it is next 
to succeed, draw QH parallel IS!. 

and Q,T perpendicular to SP, 
as also Qv parallel to the tan 
gent, and mating the diame 
ter PG in v, and the distance 




120 THE MATHEMATICAL PRINCIPLES [BOOK I. 

SP in x. Now. because of the similar triangles Pxv, SPM, and of the 
equal sides SP, SM of the one, the sides Px or Q,R and Pv of the other 
will be also equal. But (by the conic sections) the square of the ordinate 
Q,y is equal to the rectangle under the latus rectum and the segment Pv 
of the diameter ; that is (by Lem. XIII.), to the rectangle 4PS X Pv, or 
4PS X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r 
(by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q,# 2 , in 
this case, becomes equal to the rectangle 4PS X Q,R. But (because of the 
similar triangles Q#T, SPN), Q^ 2 is to QT 2 as PS 2 to SN 2 , that is (by 
Cor. 1, Lem. XIV.), as PS to SA ; that is, as 4PS X QR to 4SA x QR, 
and therefore (by Prop. IX. Lib. V., Elem.) QT* and 4SA X QR are 

SP 2 SP 2 X QT 2 

equal. Multiply these equals by ^-^-, and ^5 -will become equal 

to SP 2 X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal 
force is reciprocally as SP 2 X 4S A ; that is, because 4SA is given, recipro 
cally in the duplicate ratio of the distance SP. Q.E.I. 

COR. 1. From the three last Propositions it follows, that if any body P 
goes from the place P with any velocity in the direction of any right line 
PR, and at the same time is urged by the action of a centripetal force that 
is reciprocally proportional to the square of the distance of the places from 
the centre, the body will move in one of the conic sections, having its fo 
cus in the centre of force ; and the contrary. For the focus, the point of 
contact, and the position of the tangent, being given, a conic section may 
be described, which at that point shall have a given curvature. But the 
curvature is given from the centripetal force and velocity of the body be 
ing given ; and two orbits, mutually touching one the other, cannot be de 
scribed by the same centripetal force and the same velocity. 

COR. 2. If the velocity with which the body goes from its place P is 
such, that in any infinitely small moment of time the lineola PR may be 
thereby describe I: and the centripetal force such as in the same time to 
move the same body through the space QR ; the body will move in one of 

QT 2 . 

the conic sections, whose principal latus rectum is the quantity Tjfr in its 

ultimate state, when thelineoke PR, QR are diminished in infinitum. In 
these Corollaries I consider the circle as an ellipsis ; and I except the case 
where the body descends to the centre in a right line. 

PROPOSITION XIV. THEOREM VI. 

Tf several bodies revolve about one common centre, and the centripetal 
force is reciprocally in tlie duplicate ratio of the distance of places 
from the centre ; I say, that the principal latera recta of tfieir orbits 
are in the duplicate ratio of the areas, which the bodies by radii drawn 
to the centre describe it\ the same time. 



SEC. HI. OF NATURAL PHILOSO1 HY. 



For (by Cor 2, Prop. XIII) the latus rectum 

QT*. 

L is equal to the quantity-^-in its ultimate 




state when the points P and Q, coincide. But 
the lineola QR in a given time is as the gen 
erating centripetal force ; that is (by supposi- 

QT 2 

tion), reciprocally as SP 2 . And therefore-^-^ 

is as Q.T 2 X SP 2 ; that is, the latus rectum L is in the duplicate ratio of 
the area QT X SP. Q.E.D. 

C?OR. Hence the whole area of the ellipsis, and the rectangle under the 
axes, which is proportional to it, is in the ratio compounded of the subdu- 
plicate ratio of the latus rectum, and the ratio of the periodic time. For 
the whole area is as the area QT X SP, described in a given time, mul 
tiplied by the periodic time. 

PROPOSITION XV. THEOREM VII. 

The same things being supposed, J say, that the periodic times in ellip 
ses are in the sesquiplicate ratio of their greater axes. 
For the lesser axis is a mean proportional between the greater axis and 
the latus rectum ; and, therefore, the rectangle under the axes is in the 
ratio compounded of the subduplicate ratio of the latus rectum and the 
sesquiplicate ratio of the greater axis. But this rectangle (by Cor. o. 
Prop. XIV) is in a ratio compounded of the subduplicate ratio of the 
latus rectum, and the ratio of the periodic time. Subduct from both sides 
the subduplicate ratio of the latus rectum, and there will remain the ses 
quiplicate ratio of the greater axis, equal to the ratio of the periodic time. 
Q.E.D. 

COR. Therefore the periodic times in ellipses are the same as in circles 
whose diameters are equal to the greater axes of the ellipses. 

PROPOSITION XVI. THEOREM VIII. 

The same things being supposed, and right lines being drawn to the 
bodies that shall touch the orbits, and perpendiculars being let fall on 
those tangents from the common focus ; I say, that the velocities oj 
the bodies are in a ratio compounded of the ratio of the perpendiculars 
inversely, and the, subduplicate ratio of the principal later a recta 
direct!]). 
From the focus S draw SY perpendicular to the tangent PR, and the 

velocity of the body P will be reciprocally in the subduplicate ratio of the 

SY 2 
quantity -y . For that velocity is as the infinitely small arc PQ de- 




122 THE MATHEMATICAL PRINCIPLES [BOOK I. 

scribed in a given moment of time, that is (by 
Lem. "VII), as the tangent PR ; that is (because 
of the proportionals PR to Q,T, and SP to 

SP X Q,T 

SY), as ~y ; or as SY reciprocally, 

and SP X Q,T directly ; but SP X QT is as 
the area described in the given time, that is (by 
Prop. XIV), in the subduplicate ratio of the 
latus rectum. Q.E.D. 

COR. 1. The principal latera recta are in a ratio compounded of the 
duplicate ratio of the perpendiculars and the duplicate ratio of the ve 
locities. 

COR. 2. The velocities of bodies, in their greatest and least distances from 
the common focus, are in the ratio compounded of the ratio of the distan 
ces inversely, and the subduplicate ratio of the principal latera recta di 
rectly. For those perpendiculars are now the distances. 

COR. 3. Arid therefore the velocity in a conic section, at its greatest or 
least distance from the focus, is to the velocity in a circle, at the same dis 
tance from the centre, in the subduplicate ratio of the principal latus rec 
tum to the double of that distance. 

COR. 4. The velocities of the bodies revolving in ellipses, at their mean 
distances from the common focus, are the same as those of bodies revolving 
in circles, at the same distances ; that is (by Cor. 6. Prop. IV), recipro 
cally in the subduplicate ratio of the distances. For the perpendiculars 
are now the lesser semi-axes, and these are as mean proportionals between 
the distances and the latera recta. Let this ratio inversely be compounded 
with the subduplicate ratio of the latera recta directly, and we shall have 
the subduplicate ratio of the distance inversely. 

COR. 5. In the same figure, or even in different figures, whose principal 
latera recta are equal, the velocity of a body is reciprocally as the perpen 
dicular let fall from the focus on the tangent. 

COR. 6. In a parabola, the velocity is reciprocally in the subduplicate 
ratio.of the distance of the body from the focus of the figure; it is more 
variable in the ellipsis, and less in the hyperbola, than according to this 
ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the 
focus on the tangent of a parabola is in the subduplicate ratio of the dis 
tance. In the hyperbola the perpendicular is less variable ; in the ellipsis 
more. 

COR. 7. In a parabola, the velocity of a body at any distance from the 
focus is to the velocity of a body revolving in a circle, at the same distance 
from the centre, in the subduplicate ratio of the number 2 to 1 ; in the 
ellipsis it is less, and in the hyperbola greater, than according to this ratio, 
For (by Cor. 2 of this Prop.) the velocitv at the vertex of a parabola is ir 



SEC. III.] 



OF NATURAL PHILOSOPHY. 



123 



this ratio, and (by Cor. 6 of this Prop, and Prop. IV) the same proportion 
holds in all distances. And hence, also, in a parabola, the velocity is 
everywhere equal to the velocity of a body revolving in a circle at half the 
distance ; in the ellipsis it is less, and in the hyperbola greater. 

COR. S. The velocity of a body revolving in any conic section is to the 
velocity of a body revolving in a circle, at the distance of half the princi 
pal latus rectum of the section, as that distance to the perpendicular let 
fall from the focus on the tangent of the section. This appears from 
Cor. 5. 

COR. 9. Wherefore since (by Cor. 6, Prop. IV), the velocity of a body 
revolving in this circle is to the velocity of another body revolving in any 
other circle reciprocally in the subduplicate ratio of the distances; there 
fore, ex czqiiO) the velocity of a body revolving *in a conic section will be 
to the velocity of a body revolving* in a circle at the same distance as a 
mean proportional between that common distance, and half the principal 
latus rectum of the section, to the perpendicular let fall from the common 
focus upon the tangent of the section. 

PROPOSITION XVII. PROBLEM IX. 

Supposing the centripetal force to be reciprocally proportional to the 
squares of the distances of places from the centre, and that the abso 
lute quantity of that force is known ; it is required to determine t/te 
line which a body will describe that is let go from a given place with a 
given velocity in the direction of a given right line. 
Let the centripetal force 
tending to the point S be 
such as will make the body 
p revolve in any given orbit 
pq ; and suppose the velocity 
of this body in the place p 
is known. Then from the 
place P suppose the body P 
to be let go with a given ve 
locity in the direction of the 
line PR ; but by virtue of a 
centripetal force to be immediately turned aside from that right line into 
the conic section PQ,. This, the right line PR will therefore touch in P. 
Suppose likewise that the right line pr touches the orbit pq in p ; and if 
from S you suppose perpendiculars let fall on those tangents, the principal 
latus rectum of the conic section (by Cor. 1, Prop. XVI) will be to the 
principal latus rectum of that orbit in a ratio compounded of the duplicate 
ratio of the perpendiculars, and the duplicate ratio of the velocities ; arid 
is therefore given. Let this latus rectum be L ; the focus S of the conic 




L24 THE MATHEMATICAL PRINCIPLES [BOOK I 

section is also given. Let the angle RPH be the complement of the angle 
RPS to two right ; and the line PH, in which the other focus II is placed, 
is given by position. Let fall SK perpendicular on PH, and erect the 
conjugate semi-axis BC ; this done, we shall have SP 2 2KPH + PH 2 
= SH 2 = 4CH 2 = 4BH 2 4BC 2 = SP + PH 2 L X SiM 



SP 2 + 2SPH + PH 2 L x SP + PH. Add on both sides 2KPH 



SP 2 PH 2 + L X SP + PH, and we shall have L X SP + PH = 2SPH 
f 2KPH, or SP + PH to PH, as 2SP + 2KP to L. Whence PH is 
given both in length and position. That is, if the velocity of the body 
in P is such that the latus rectum L is less than 2SP + 2KP, PH will 
lie on the same side of the tangent PR w r ith the line SP ; and therefore 
the figure will be an ellipsis, which from the given foci S, H, and the 
principal axis SP + PH, is given also. But if the velocity of the body 
is so great, that the latus rectum L becomes equal to 2SP + 2KP, the 
length PH will be infinite ; and therefore, the figure will be a parabola, 
which has its axis SH parallel to the line PK, and is thence given. But 
if the body goes from its place P with a yet greater velocity, the length 
PH is to be taken on the other side the tangent ; and so the tangent pas 
sing between the foci, the figure will be an hyperbola having its principal 
axis equal to the difference of the lines SP and PH, and thence is given. 
Por if the body, in these cases, revolves in a conic section so found, it is 
demonstrated in Prop. XI, XII, and XIII, that the centripetal force will 
be reciprocally as the square of the distance of the body from the centre 
of force S ; and therefore we have rightly determined the line PQ,, which 
a body let go from a given place P with a given velocity, and in the di 
rection of the right line PR given by position, would describe with such a 
force. Q.E.F. 

COR. 1. Hence in every conic section, from the principal vertex D, the 
latus rectum L, and the focus S given, the other focus H is given, by 
taking DH to DS as the latus rectum to the difference between the latus 
rectum and 4US. For the proportion, SP + PH to PH as 2SP + 2KP 
to L, becomes, in the case of this Corollary, DS + DH to DH as 4DS to 
L, and by division DS to DH as 4DS L to L. 

COR. 2. Whence if the velocity of a body in the principal vertex D ig 
given, the orbit may be readily found ; to wit, by taking its latus rectum 
to twice the distance DS, in the duplicate ratio of this given velocity to 
the velocity of a body revolving in a circle at the distance DS (by Cor. 
3, Prop. XVI.), and then taking DH to DS as the latus rectum to the 
difference between the latus rectum and 4DS. 

COR. 3. Hence also if a body move in any conic section, and is forced 
out of its orbit by any impulse, you may discover the orbit in which it will 
afterwards pursue its Bourse. For bv compounding the proper motion oi 



SEC. IV.] OF NATURAL PHILOSOPHY. 125 

the body with that motion, which the impulse alone would generate, you 
will have the motion with which the body will go off from a given place 
of impulse in the direction of a right line given in position. 

COR. 4. And if that body is continually disturbed by the action of some 
foreign force, we may nearly know its course, by collecting the changes 
which that force introduces in some points, and estimating the continual 
changes it will undergo in the intermediate places, from the analogy that 
appears in the progress of the series. 

SCHOLIUM. 

If a body P, by means of a centripetal 
force tending to any given point R, move 
in the perimeter of any given conic sec 
tion whose centre is C ; and the law of 
the centripetal force is required : draw 
CG parallel to the radius RP, and meet 
ing the tangent PG of the orbit in G ; 
and the force required (by Cor. 1, and 

CG 3 
Schol. Prop. X., and Cor. 3, Prop. VII.) will be as - 




SECTION IV. 

Of the finding of elliptic, parabolic, and hyperbolic orbits, from ttu. 
focus given. 

LEMMA XV. 

If from the two foci S, II, of any ellipsis or hyberbola, we draw to any 
third point V the right lines SV, H V, whereof one HV is equal to the 
principal axis of the figure, thai is, to the axis in which the foci are 
situated, the other, SV, is bisected in T by t/ie perpendicular TR let 
fall upon it ; that perpendicular TR will somewhere touch the conic 
section : and, vice versa, if it does touch it, HV will be equal to the 
principal axis of the figure. 
For, let the perpendicular TR cut the right line 
HV, produced, if need be, in R ; and join SR. Be 
cause TS, TV are equal, therefore the right lines SR, 
VR, as well as the angles TRS, TRV, will be also 
equal. Whence the point R will be in the conic section, and the perpen 
dicular TR will touch the same ; and the contrary. Q.E.D. 




126 THE MATHEMATICAL PBINCIP, -ES [BOOK 1 

PROPOSITION XVIII. PROBLEM X. 

From a focus and the principal axes given, to describe elliptic and hy 
perbolic trajectories, which shall pass through given points, and touch 
right lines given by position. 
Let S be the common focus of the figures ; AB A 33 

the length of the principal axis of any trajectory ; r p T~* 

P a point through which the trajectory should \ /R 

pass ; and TR a right line which it should touch. / \ 



About the centre P, with the interval AB SP, \ S ~~yf 

if the orbit is an ellipsis, or AB {- SP, if the y> G ^ 

orbit is an hyperbola, describe the circle HG. On the tangent TR let fall 
the perpendicular ST, and produce the same to V, so that TV may be 
equal to ST; and about V as a centre with the interval AB describe the 
circle FH. In this manner, whether two points P, p, are given, or two 
tangents TR, tr, or a point P and a tangent TR, we are to describe two 
circles. Let H be their common intersection, and from the foci S, H, with 
the given axis describe the trajectory : I say, the thing is done. For (be 
cause PH -f- SP in the ellipsis, and PH SP in the hyperbola, is equal 
to the axis) the described trajectory will pass through the point P, and (by 
the preceding Lemma) will touch the right line TR. And by the same 
argument it will either pass through the two points P, p, or touch the two 
right lines TR, tr. Q.E.F. 

PROPOSITION XIX. PROBLEM XI. 

About a given focus, to describe a parabolic trajectory, which shall pass 
through given points, and touch right lines given by position. 
Let S be the focus, P a point, and TR a tangent of 
the trajectory to be described. About P as a centre, 
with the interval PS, describe the circle FG. From 
the focus let fall ST perpendicular on the tangent, and 
produce the same to V, so as TV may be equal to ST. 
After the same manner another circle fg is to be de 
scribed, if another point p is given ; or another point v 
is to be found, if another tangent tr is given; then draw 
the right line IF, which shall touch the two circles YG,fg, if two points 
P, p are given ; or pass through the two points V, v, if two tangents TR, 
tr, are given : or touch the circle FG, and pass through the point V, if the 
point P and the tangent TR are given. On FI let fall the perpendicular 
SI, and bisect the same in K ; and with the axis SK and principal vertex K 
describe a parabola : I say the thing is done. For this parabola (because 
SK is equal to IK, and SP to FP) will pass through the point P ; and 




/KS 




SEC. IV.] OF NATURAL PHILOSOPHY. 127 

(by Cor. 3, Lem. XIV) because ST is equal to TV. and STR a light an 
gle, it will touch the right line TR. Q.E.F. 

PROPOSITION XX. PROBLEM XII. 

About a given focus to describe any trajectory given in specie which shah 
pass through given points, and touch right lines given by position. 
CASE 1. About the focus S it is re- 
uired to describe a trajectory ABC, pass 
ing through two points B, C. Because the 
trajectory is given in specie, the ratio of the 

principal axis to the distance of the foci GAS H 

will be given. In that ratio take KB to BS, and LC to CS. About the 
centres B, C, with the intervals BK, CL, describe two circles ; and on the 
right line KL, that touches the same in K and L, let fall the perpendicu 
lar SG ; which cut in A and a, so that GA may be to AS, and Ga to aS, 
as KB to BS ; and with the axis A., and vertices A, a, describe a trajectory : 
I say the thing is done. For let H be the other focus of the described 
figure, and seeing GA is to AS as Ga to aS, then by division we shall 
have Ga GA, or Aa to S AS, or SH in the same ratio ; and therefore 
in the ratio which the principal axis of the figure to be described has to 
the distance of its foci ; and therefore the described figure is of the same 
species with the figure which was to be described. And since KB to BS, 
and LC to CS, are in the same ratio, this figure will pass through tht- 
points B, C, as is manifest from the conic sections. 

CASE 2. About the focus S it is required to 
describe a trajectory which shall somewhere 
touch two right lines TR, tr. From the focus 
on those tangents let fall the perpendiculars 
ST, St, which produce to V, v, so that TV, tv 
may be equal to TS, tS. Bisect Vv in O, and j 
erect the indefinite perpendicular OH, and cut I. 
the right line VS infinitely produced in K and V 
k, so that VK be to KS, and VA* to A~S, as the principal axis of the tra 
jectory to be described is to the distance of its foci. On the diameter 
K/J describe a circle cutting OH in H ; and with the foci S, H, and 
principal axis equal to VH, describe a trajectory : I say, the thing is done. 
For bisecting Kk in X, and joining HX, HS, HV, Hv, because VK is to 
KS as VA- to A*S ; and by composition, as VK -f- V/c to KS + kS ; and 
by division, as VA* VK to kS KS, that is, as 2VX to 2KX, and 
2KX to 2SX, and therefore as VX to HX and HX to SX, the triangles 
VXH, HXS will be similar ; therefore VH will be to SH as VX to XH ; 
and therefore as VK to KS. Wherefore VH, the principal axis of the 
described trajectory, has the same ratio to SH, the distance of the foci, as 




12S 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1. 



K S 



the principal axis of the trajectory which was to be described has to the 
distance of its foci ; and is therefore of the same species. Arid seeing VH, 
vH are equal to the principal axis, and VS, vS are perpendicularly bisected 
by the right lines TR, tr, it is evident (by Lem. XV) that those right 
lines touch the described trajectory. Q,.E.F. 

CASE. 3. About the focus S it is required to describe a trajectory, which 
shall touch a right line TR in a given Point R. On the right line TR 
Jet fall the perpendicular ST, which produce to V, so that TV may be 
equal to ST ; join VR, and cut the right line VS indefinitely produced 
in K and k, so. that VK may be to SK, and V& to SAr, as the principal 
axis of the ellipsis to be described to the distance of its foci ; and on the 
diameter KA: describing a circle, cut the H 

right line VR produced in H ; then with 
the foci S, H, and principal axis equal to R 
VH, describe a trajectory : I say, the thing .--- 
is done. For VH is to SH as VK to SK, V" "1 
and therefore as the principal axis of the trajectory which was to be de 
scribed to the distance of its foci (as appears from what we have demon 
strated in Case 2) ; and therefore the described trajectory is of the same 
species with that which was to be described ; but that the right line TR, 
by which the angle VRS is bisected, touches the trajectory in the point R, 
is certain from the properties of the conic sections. Q.E.F. 

CASE 4. About the focus S it is r 

required to describe a trajectory 
APB that shall touch a right line 
TR, and pass through any given 
point P without the tangent, and 
shall be similar to the figure apb, 
described with the principal axis ab, 
and foci s, h. On the tangent TR 
let fall the perpendicular ST, which / .. ,.---" " 
produce to V, so that TV may be 
equal to ST ; and making the an 
gles hsq, shq, equal to the angles VSP, SVP, about q as a centre, and 
with an interval which shall be to ab as SP to VS, describe a circle cut 
ting the figure apb in p : join sp, and draw 
SH such that it may be to sh as SP is to sp, 
and may make the angle PSH equal to the 
angle psh, and the angle VSH equal to the 
angle pyq. Then with the foci S, H, and B 
principal axis AB, equal to the distance VH, 
describe a conic section : I say, the thing is 
done ; for if sv is drawn so that it shall be to 





SEC. IV.] OF NATURAL PHILOSOPHY. 129 

sp as sh is to sq, and shall make the angle vsp equal to the angle hsq, and 
the angle vsh equal to the angle psq, the triangles svh, spq, will be similar, 
and therefore vh will be to pq as sh is to sq ; that is (because of the simi 
lar triangles VSP, hsq), as VS is to SP ? or as ab to pq. Wherefore 
vh and ab are equal. But, because of the similar triangles VSH, vsh, VH 
is to SH as vh to sh ; that is, the axis of the conic section now described 
is to the distance of its foci as the axis ab to the distance of the foci sh ; 
and therefore the figure now described is similar to the figure aph. But, 
because the triangle PSH is similar to the triangle psh, this figure passes 
through the point P ; and because VH is equal to its axis, and VS is per 
pendicularly bisected by the rght line TR, the said figure touches the 
right line TR. Q.E.F. 

LEMMA XVI. 

From three given points to draw to afonrth point that is not given three 
right lines whose differences shall be either given, or none at all. 
CASE 1. Let the given points be A, B, C, and Z the fourth point which 
we are to find ; because of the given difference of the lines AZ, BZ, the 
locus of the point Z will be an hyperbola 
whose foci are A and B, and whose princi 
pal axis is the given difference. Let that 
axis be MN. Taking PM to MA as MN 
is to AB, erect PR perpendicular to AB, 
and let fall ZR perpendicular to PR ; then 
from the nature of the hyperbola, ZR will 
be to AZ as MN is to AB. And by the 
like argument, the locus of the point Z will 
be another hyperbola, whose foci are A, C, and whose principal axis is the 
difference between AZ and CZ ; and QS a perpendicular on AC may be 
drawn, to which (QS) if from any point Z of this hyperbola a perpendicular 
ZS is let fall (this ZS), shall be to AZ as the difference between AZ and 
CZ is to AC. Wherefore the ratios of ZR and ZS to AZ are given, and 
consequently the ratio of ZR to ZS one to the other ; and therefore if the 
right lines RP, SQ, meet in T, and TZ and TA are drawn, the figure 
TRZS will be given in specie, and the right line TZ, in which the point 
Z is somewhere placed, will be given in position. There will be given 
also the right line TA, and the angle ATZ ; and because the ratios of AZ 
and TZ to ZS are given, their ratio to each other is given also ; and 
thence will be given likewise the triangle ATZ, whose vertex is the point 
Z. Q.E.I. 

CASE 2. If two of the three lines, for example AZ and BZ, are equal, 
draw the right line TZ so as to bisect the right line AB ; then find the 
triangle ATZ as above. Q.E.I. 




130 



THE MATHEMATICAL PRINCIPLES 



[BOOK I. 




CASE 3. If all the three are equal, the point Z will be placed in the 
centre of a circle that passes through the points A, B, C. Q.E.I. 

This problematic Lemma is likewise solved in Apollonius s Book oi 
Tactions restored by Vieta. 

PROPOSITION XXL PROBLEM XIII. 

About a given focus to describe a trajectory that shall pass through 
given points and touch right Hues given by position. 
Let the focus S, the point P, and the tangent TR be given, and suppose 

that the other focus H is to be found. 

On the tangent let fall the perpendicular 

ST, which produce to Y, so that TY may 

be equal to ST, and YH will be equal 

to the principal axis. Join SP, HP, and 

SP will be the difference between HP and 

the principal axis. After this manner, 

if more tangents TR are given, or more 

points P. we shall always determine as 

many lines YH, or PH, drawn from the said points Y or P, to the focus 

H, which either shall be equal to the axes, or differ from the axes by given 

lengths SP ; and therefore which shall either be equal among themselves, 

or shall have given differences ; from whence (by the preceding Lemma). 

that other focus H is given. But having the foci and the length of the 

axis (which is either YH, or, if the trajectory be an ellipsis, PH -f SP ; 

or PH SP, if it be an hyperbola), the trajectory is given. Q.E.I. 

SCHOLIUM. 

When the trajectory is an hyperbola, I do not comprehend its conjugate 
hyperbola under the name of tins trajectory. For a body going on with a 
continued motion can never pass out of one hyperbola into its conjugate 
hyperbola. 

The case when three points are given 
is more readily solved thus. Let B, C, 
I), be the given points. Join BC, CD, 
and produce them to E, F, so as EB may 
be to EC as SB to SC ; and FC to FD 
as SC to SD. On EF drawn and pro 
duced let fall the perpendiculars SG, 
BH, and in GS produced indefinitely E 
take GA to AS, and Ga to aS, as HB 
is to BS ; then A will be the vertex, and Aa the principal axis of the tra 
jectory ; which, according as GA is greater than, equal to, or less than 




SEC. V.] OF NATURAL PHILOSOPHY. 131 

AS. will be either an ellipsis, a parabola, or an hyperbola ; the point a in 
the first case falling on the same side of the line GP as the point A ; in 
the second, going oft* to an infinite distance ; in the third, falling on the 
other side of the line GP. For if on GF the perpendiculars CI, DK are 
let fall, TC will be to HB as EC to EB ; that is, as SO to SB ; and by 
permutation, 1C to SC as HB to SB, or as GA to SA. And, by the like 
argument, we may prove that KD is to SD in the same ratio. Where 
fore the points B, C, D lie in a conic section described about the focus S, 
in such manner that all the right lines drawn from the focus S to the 
several points of the section, and the perpendiculars let fall from the same 
points on the right line GF, are in that given ratio. 

That excellent geometer M. De la Hire has solved this Problem much 
after the same way, in his Conies, Prop. XXV., Lib. VIII. 



SECTION V. 

How the orbits are to be found when neither focus is given. 

LEMMA XVII. 

If from any point P of a given conic section, to the four produced sides 
AB, CD, AC, DB, of any trapezium ABDC inscribed in that section, 
as many right lines PQ, PR, PS, PT are drawn in given ang 7 ei, 
each line to each side ; the rectangle PQ, X PR of those on the opposite 
sides AB, CD, will be to the rectangle PS X PT of those on tie other 
two opposite sides AC, BD, in a given ratio. 
CASE 1. Let us suppose, first, that the lines drawn 

to one pair of opposite sides are parallel to either of I ^^ p ; T 
the other sides ; as PQ and PR to the side AC, and s | 
PS and PT to the side AB. And farther, that one 
pair of the opposite sides, as AC and BD, are parallel 
betwixt themselves; then the right line which bisects^ IQ I3 

those parallel sides will be one of the diameters of the 1L 

conic section, and will likewise bisect RQ. Let O be the point in which 
RQ is bisected, and PO will be an ordinate to that diameter. Produce 
PO to K, so that OK may be equal to PO, and OK will be an ordinate 
on the other side of that diameter. Since, therefore, the points A, B ; P 
and K are placed in the conic section, and PK cuts AB in a given angle, 
the rectangle PQK (by Prop. XVII., XIX., XXI. and XXI1L, Book III., 
of Apollonius s Conies) will be to the rectangle AQB in a given ratio. 
But QK and PR are equal, as being the differences of the equal lines OK, 
OP, and OQ, OR ; whence the rectangles PQK and PQ X PR are equal ; 
and therefore the rectangle PQ X PR is to the rectangle A^ B, that Is, to 
the rectangle PS X PT in a given ratio. Q.E.D 




132 



THE MATHEMATICAL PRINCIPLES 



[BOOK I 





CASE 2. Let us next suppose that the oppo 
site sides AC and BD of the trapezium are not 
parallel. Draw Be/ parallel to AC, and meeting 
as well the right line ST in /, as the conic section 
in d. Join Cd cutting PQ in r, and draw DM 
parallel to PQ, cutting Cd in M, and AB in N. 

Then (because of the similar triangles BTt, 

DBN), Et or PQ is to Tt as DN to NB. And ^^ Q N 

so Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antece- 
dents by the antecedents, and the consequents by the consequents, as the 
rectangle PQ X Rr is to the rectangle PS X Tt, so will the rectangle 
N i)M be to the rectangle ANB ; and (by Case 1) so is the rectangle 
PQ X Pr to the rectangle PS X Pt : and by division, so is the rectangle 
PQ X PR to the rectangle PS X PT. Q.E.D. 

CASE 3. Let us suppose, lastly, the four lines 
?Q, PR, PS, PT, not to be parallel to the sides 
AC, AB, but any way inclined to them. In their 
place draw Pq, Pr, parallel to AC ; and Ps, Pt 
parallel to AB ; and because the angles of the 
triangles PQ</, PRr, PSs, PTt are given, the ra- 
tios of IQ to Pq, PR to Pr, PS to P*, PT to Pt 
will b? also given; and therefore the compound 
ed ratios Pk X PR to P? X Pr, and PS X PT to Ps X Pt are 
given. But from what we have demonstrated before, the ratio of Pq X Pi 
to Ps X Pt is given ; and therefore also the ratio of PQ X PR to PS X 
PT. Q.E.D. 

LEMMA XVIII. 

The s niL things supposed, if the rectangle PQ X PR of the lines drawn 
to the two opposite sides of the trapezium is to the rectangle PS X PT 
of those drawn to the other two sides in a given ratio, the point P, 
from whence those lines are drawn, will be placed in a conic section 
described about the trapezium. 
Conceive a conic section to be described pas 
sing through the points A, B, C, D, and any 

one of the infinite number of points P, as for 

example p ; I say, the point P will be always c 1 

placed in this section. If you deny the thing, 

join AP cutting this conic section somewhere 

else, if possible, than in P, as in b. Therefore 

if from those points p and b, in the given angles ^ B 

to the sides of the trapezium, we draw the right 

lines pq, pr, ps, pt, and bk, bn, bf, bd, we shall have, as bk X bn to bf X bd, 




SEC. V.] OF NATURAL PHILOSOPHY 133 

so (by Lem. XVII) pq X pr to ps X pt ; and so (by supposition) PQ x 
PR to PS X PT. And because of the similar trapezia bkAf, PQAS, as 
bk to bf, so PQ to PS. Wherefore by dividing the terms of the preceding 
proportion by the correspondent terms of this, we shall have bn to bd as 
PR to PT. And therefore the equiangular trapezia ~Dnbd, DRPT, are 
similar, and consequently their diagonals D6, DP do coincide. Wherefore 
b falls in the intersection of the right lines AP, DP, and consequently 
coincides with the point P. And therefore the point P, wherever it is 
taken, falls to be in the assigned conic section. Q.E.D. 

COR. Hence if three right lines PQ, PR, PS, are drawn from a com 
mon point P, to as many other right lines given in position, AB, CD, AC, 
each to each, in as many angles respectively given, and the rectangle PQ 
X PR under any two of the lines drawn be to the square of the third PS 
in a given ratio ; the point P, from which the right lines are drawn, will 
be placed in a conic section that touches the lines AB ; CD in A and C 
and the contrary. For the position of the three right lines AB, CD, AC 
remaining the same, let the line BD approach to and coincide with the 
line AC ; then let the line PT come likewise to coincide with the line PS ; 
and the rectangle PS X PT will become PS 2 , and the right lines AB, CD, 
which before did cut the curve in the points A and B, C and D, can no 
(onger cut, but only touch, the curve in those coinciding points. 

SCHOLIUM. 

In this Lemma, the name of conic section is to be understood in a large 
sense, comprehending as well the rectilinear section through the vertex of 
the cone, as the circular one parallel to the base. For if the point p hap 
pens to be in a right line, by which the points A and D, or C and B are 
joined, the conic section will be changed into two right lines, one of which 
is that right line upon which the point p falls, 
and the other is a right line that joins the other 
two of *he four points. If the two opposite an 
gles of the trapezium taken together are equal c 
to two right angles, and if the four lines PQ, 
PR, PS, PT, are drawn to the sides thereof at 
right angles, or any other equal angles, and the 
rectangle PQ X PR under two of the lines 
drawn PQ and PR, is equal to the rectangle 
PS X PT under the other two PS and PT, the conic section will become 
a circle. And the same thing will happen if the four lines are drawn in 
any angles, and the rectangle PQ X PR, under one pair of the lines drawn, 
is to the rectangle PS X PT under the other pair as the rectangle under 
the sines of the angles S, T, in which the two last lines PS, PT are drawn 
to the rectangle under the sines of the angles Q, R, in which the first tw 




134 THE MATHEMATICAL PRINCIPLES [BOOK 1. 

PQ, PR are drawn. In all other cases the locus of the point P will be 
one of the three figures which pass commonly by the name of the conic 
sections. But in room of the trapezium A BCD, we may substitute a 
quadrilateral figure whose two opposite sides cross one another like diago 
nals. And one or two of the four points A, B, C, D may be supposed to 
be removed to an infinite distance, by which means the sides of the figure 
which converge to those points, will become parallel ; and in this case the 
conic section will pass through the other points, and will go the same way 
as the parallels in, infinitum. 

LEMMA XIX. 

To find a point P from which if four right lines PQ, PR, PS, PT an 
drawn to as many other right lines AB, CD, AC, BD, given by posi 
tion, each to each, at given angles, the rectangle PQ X PR, under any 
two of the lines drawn, shall be to the rectangle PS X PT, under the 
other tivo. in a given ratio. 
Suppose the lines AB, CD, to which the two 
right lines PQ, PR, containing one of the rect 
angles, are drawn to meet two other lines, given 
by position, in the points A, B, C, D. From one 
of those, as A, draw any right line AH, in which 
you would find the point P. Let this cut the 
opposite lines BD, CD, in H and I ; and, because 
all the angles of the figure are given, the ratio of 
PQ to PA, and PA to PS, and therefore of PQ 

to PS, will be also given. Subducting this ratio from the given ratio oi 
PQ X PR to PS X PT, the ratio of PR to PT will be given ; and ad 
ding the given ratios of PI to PR, and PT to PH, the ratio of PI to PH. 
and therefore the point P will be given. Q.E.I. 

COR. 1. Hence also a tangent may be drawn to any point D of the 
locus of all the points P. For the chord PD, where the points P and D 
meet, that is, where AH is drawn through the point D, becomes a tangent. 
In which case the ultimate ratio of the evanescent lines IP and PH will 
be found as above. Therefore draw CF parallel to AD, meeting BD in 
F, and cut it in E in the same ultimate ratio, then DE will be the tan 
gent ; because CF and the evanescent IH are parallel, and similarly cut in 
E and P. 

COR. 2. Hence also the locus of all the points P may be determined. 
Through any of the points A, B, C, D, as A, draw AE touching the locus, 
and through any other point B parallel to the tangent, draw BF meeting 
the locus in F ; and find the point F by this Lemma. Bisect BF in G, 
and, drawing the indefinite line AG, this will be the position of the dia 
meter to which BG and FG are ordinates. Let this AG meet the locus 




SEC. V.J 



OF NATURAL PHILOSOPHY. 



in H, and AH will be its diameter or latus trans- 
versum. to which the latus rectum will be as BG 2 
to AG X GH. If AG nowhere meets the locus, 
the line AH being infinite, the locus will be a par 
abola ; and its latus rectum corresponding to the 



diameter AG will be -.-7^ 
AC* 



But if it does meet it 




anywhere, the locus will be an hyperbola, when 

the points A and H are placed on the same side the point G ; and an 
ellipsis, if the point G falls between the points A and H ; unless, perhaps, 
the angle AGB is a right angle, and at the same time BG 2 equal to the 
rectangle AGH, in which case the locus will be a circle. 

And so we have given in this Corollary a solution of that famous Prob 
lem of the ancients concerning four lines, begun by Euclid, and carried on 
by Apollonius ; and this not an analytical calculus, but a geometrical com 
position, such as the ancients required. 

LEMMA XX. 

If the two opposite angular points A and P of any parallelogram ASPQ 
touch any conic section in the points A and P ; and the sides AQ, AS 
of one of those angles, indefinitely produced, meet the same conic section 
in B and C ; and from the points of concourse, B and C to any fifth 
point D of the conic section, two right lines BD, CD are drawn meet- 
ing tlie two other sides PS, PQ of the parallelogram, indefinitely pro 
duced in T and R ; the parts PR and PT, cut off from the sides, will 
always be one to the other in a given ratio. And vice versa, if those 
parts cut off are one to the other in a given ratio, the locus of the point 
D will be a conic section passing through the four points A, B, C, F 
CASE 1. Join BP, CP, and from the point 

D draw the two right lines DG, DE, of which 

the first DG shall be parallel to AB, and 

meet PB, PQ, CA in H, I, G ; and the other 

DE shall be parallel to AC, and meet PC, 

PS, AB, in F, K, E ; and (by Lem. XVII) 

the rectangle DE X DF will be to the rect 

angle DG X DH in a given ratio. But 

PQ is to DE (or IQ) as PB to HB, and con 

sequently as PT to DH ; and by permutation PQ, is to PT as DE to 

DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS 

to DG ; and by permutation PR is to PS as DF to DG ; and, by com 

pounding those ratios, the rectangle PQ X PR will be to the rectangle 

PS X PT as the rectangle DE X DF is to the rectangle DG X DH. 

and consequently in "a given ratio. But PQ and PS are given, and there 

fore the ratio of PR to PT is given. Q.E.D. 




136 



THE MATHEMATICAL PRINCIPLES 



CASE 2. But if PR and PT are supposed to be in a given ratio one to 
the other, then by going back again, by a like reasoning, it will follow 
that the rectangle DE X DF is to the rectangle DG X DH in a given 
rati) ; and so the point D (by Lem. XVIII) will lie in a conic section pass 
ing through the points A., B, C, P, as its locus. Q.E.I). 

COR. 1. Hence if we draw BC cutting PQ in r and in PT take Pt to 
Pr in the same ratio which PT has to PR ; then Et will touch the conic 
section in the point B. For suppose the point D to coalesce with the point 
B, so that the chord BD vanishing, BT shall become a tangent, and CD 
and BT will coincide with CB and Bt. 

COR. 2. And, vice versa, if Bt is a tangent, and the lines BD, CD meet 
in any point D of a conic section, PR will be to PT as Pr to Pt. And, 
on the contrary, if PR is to PT as Pr to Pt, then BD and CD will meet 
in some point D of a conic section. 

COR. 3. One conic section cannot cut another conic section in more than 
four points. For, if it is possible, let two conic sections pass through the 
h ve points A, B, C, P, O ; and let the right line BD cut them in the 
points D, d, and the right line Cd cut the right line PQ, in q. Therefore 
PR is to PT as Pq to PT : whence PR and Pq are equal one to the other, 
against the supposition. 

LEMMA XXI. 

If two moveable and indefinite right lines BM, CM drawn through given 
points B, C, as poles, do by their point of concourse M describe a third 
right line MN given by position ; and other two indefinite right lines 
BD,CD are drawn, making with the former two at those given points 
B, C, given angles, MBD, MCD : I say, that those two right lines BD, 
CD will by their point of concourse D describe a conic section passing 
through the points B, C. And, vice versa, if the right lints BD, CD 
do by their point of concourse D describe a conic section passing 
through the given points B, C, A, and the angle DBM is always 
equal to the giren angle ABC, as well as the angle DCM always 
equal to the given angle ACB, the point M will lie in a right line 
given by position, as its locus. 
For in the right line MN let a point 

N be given, and when the moveable point 

M falls on the immoveable point N. let 

the moveable point D fall on an immo 
vable point P. Join ON, BN, CP, BP, 

and from the point P draw the right lines 

PT, PR meeting BD, CD in T and R, C 

and making the angle BPT c jual to the 

given angle BNM, and the angle CPR 




SEC. V.J 



OF NATURAL PHILOSOPHY. 



137 




equal to the given angle CNM. Wherefore since (by supposition) the an 
gles MBD, NBP are equal, as also the angles MOD, NCP, take away the 
angles NBD and NOD that are common, and there will remain the angles 
NBM and PBT, NCM and PCR equal; and therefore the triangles NBM, 
PBT are similar, as also the triangles NCM, PCR. Wherefore PT is to 
NM as PB to NB ; and PR to NM as PC to NC. But the points, B, C, 
N, P are immovable: wheiefore PT and PR have a given ratio to NM, 
and consequently a given ratio between themselves; and therefore, (by 
Lemma XX) the point D wherein the moveable right lines BT and CR 
perpetually concur, will be placed in a conic section passing through the 
points B. C, P. Q.E.D. 

And, vice versa, if the moveable point 
D lies in a conic section passing through 
the given points B, C, A ; and the angle 
DBM is always equal to the given an 
gle ABC, and the angle DCM always 
equal to the given angle ACB, and when 
the point D falls successively on any 
two immovable points p, P, of the conic 
section, the moveable point M falls suc 
cessively on two immovable points /?, N. 
Through these points ??, N, draw the right line nN : this line nN will be 
the perpetual locus of that moveable point M. For, if possible, let the 
point M be placed in any curve line. Therefore the point D will be placed 
in a conic section passing through the five points B, C, A, p, P, when the 
point M is perpetually placed in a curve line. But from what was de 
monstrated before, the point D will be also placed in a conic section pass 
ing through the same five points B, C, A, p, P, when the point M is per 
petually placed in a right line. Wherefore the two conic sections will both 
pass through the same five points, against Corol. 3, Lem. XX. It is 
therefore absurd to suppose that the point M is placed in a curve line. 
QE.D. 

PROPOSITION XXII. PROBLEM XIV. 

To describe a trajectory that shall pass through Jive given points. 
Let the five given points be A, B, C, P, D. c 
From any one of them, as A, to any other s v 
two as B, C, which may be called the poles, 
draw the right lines AB, AC, and parallel to 
those the lines TPS, PRO, through the fourth 
point P. Then from the two poles B, C, 
draw through the fifth point D two indefinite 
lines BDT, CRD, meeting with the last drawn lines TPS, PRQ (the 




138 



THE MATHEMATICAL PRINCIPLES 



IBOOK L 




former with the former, and the latter with the latter) in T and R. Then 
drawing the right line tr parallel to TR, cutting off from the right lines 
PT, PR, any segments Pt, Pr, proportional to PT, PR ; and if through 
their extremities, t, r, and the poles B, C, the right lines lit, Cr are drawn, 
meeting in d, that point d will be placed in the trajectory required. For 
(by Lena. XX) that point d is placed in a conic section passing through 
the four points A, B, C, P ; and the lines R/ , TV vanishing, the point d 
comes to coincide with the point D. Wherefore the conic section passes 
through the five points A, B, C, P, D. Q.E.D. 

The same otherwise. 

Of the given points join any three, as A, B, 
C ; and about two of them 15, C, as poles, 
making the angles ABC, ACB of a given 
magnitude to revolve, apply the legs BA, 
CA, first to the point D, then to the point P, 
and mark the points M, N, in which the other 
legs BL, CL intersect each other in both cases. C 
Draw the indefinite right line MN, and let 
those moveable angles revolve about their 
poles B, C, in such manner that the intersection, which is now supposed to 
be ???, of the legs BL, CL ; or BM 7 CM, may always fall in that indefinite 
right line MN ; and the intersection, which is now supposed to be d, of the 
legs BA ^A, or BD ; CD, will describe the trajectory required, PADc/B. 
For (by Lem. XXI) the point d will be placed in a conic section passing- 
through the points B, C ; and when the point m comes to coincide with 
the points L, M, N, the point d will (by construction) come to coin 
cide with the points A, D, P. Wherefore a conic section will be described 
that shall pass through the five points A, B. C, P, D. Q,.E.F. 

COR. 1. Hence a right line may be readily drawn which shall be a tan 
gent to the trajectory in any given point B. Let the point d come to co 
incide with the point B, arid the right line Bt/ Avill become the tangent 
required. 

COR. 2. Hence also may be found the centres, diameters, and latera recta 
of the trajectories, as in Cor. 2, Lem. XIX. 

SCHOLIUM. 

The former of these constructions will be- c 
come something more simple by joining , 
and in that line, produced, if need be, aking 
Bp to BP as PR is to PT ; and t rough p 
draw the indefinite right inc j0e parallel to S 
PT, and in that line pe taking always pe 
equal to Pi , and draw the right lines Be, Cr 




SEC. Y.J 



OF NATURAL PHILOSOPHY. 



139 




to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in 

the same ratio, pe and Pr will be always equal. After this manner the 
points of the trajectory are most readily found, unless you would rather 
describe the curve mechanically, as in the second construction. 

PROPOSITION XXIII. PROBLEM XV. 

To describe a trajectory that shall pass through four given points, and 

touch a right line given by position. 

CASE 1. Suppose that HB is the 
given tangent, B the point of contact, 
and C, 1., P, the three other given 
points. Jo n BC. and draw IS paral 
lel to BH, and PQ parallel to BC ; 
complete the parallelogram BSPQ. 
Draw BD cutting SP in T, and CD 
cutting PQ, in R. Lastly, draw any 
line tr parallel to TR, cutting off 
from PQ, PS, the segments Pr, Pt proportional to PR, PT respectively ; 
and draw Cr, Bt their point of concourse d will (by Lem. XX) always fall 
on the trajectory to be described. 

The same otherwise. 

1 et tl e angle CBH of a given magnitude re 
volve about the pole B ; as also the rectilinear ra- 
d : us 1C, both ways produced, about the pole C. 
Mark the points M, N, on which the leg BC of 
the angle cuts that radius when BH ; the other 
leg thereof, meets the same radius in the points 
P and D. Then drawing the indefinite line MN, 
let that radius CP or CD and the leg BC of the 
angle perpetually meet in this Ikie; and the 
point of concourse of the other leg BH with the 
radius will delineate the trajectory required. 

For if in the constructions of the preceding Problem the point A comes 
to a coincidence with the point B, the lines CA and CB will coincide, and 
the line AB, in its last situation, will become the tangent BH ; and there 
fore the constructions there set down will become the same with the con 
structions here described. Wherefore the concourse of the leg BH with 
the radius will describe a conic section passing through the points C, D, 
P, and touching the line BH in the point B. Q.E.F. 

CASE 2. Suppose the four points B, C, D, P, given, being situated with- 
ont the tangent HI. Join each two by the lines BD, CP meeting in G, 
and cutting the tangent in H and I. Cut the tangent in A in such mannr: 




140 



THE MATHEMATICAL PRINCIPLES 



[BOOK I 




X 



IT 



that HA may be to IA as the rectangle un 
der a mean proportional between CG and 
GP, and a mean proportional between BH 
and HD is to a rectangle under a mean pro 
portional between GD and GB, and a mean 
proportional betweeen PI and 1C, and A will 
be the point of contact. For if HX, a par 
allel to the right line PI, cuts the trajectory 
in any points X and Y, the point A (by the 
properties of the conic sections) will come to be so placed, that HA 2 will 
become to AP in a ratio that is compounded out of the ratio of the rec 
tangle XHY to the rectangle BHD, or of the rectangle CGP to the rec 
tangle DGB; and the ratio of the rectangle BHD to the rectangle PIC. 
But after the point of contac.t A is found, the trajectory will be described as 
in the first Case. Q.E.F. But the point A may be taken either between 
or without the points H and I, upon which account a twofold trajectory 
may be described. 

PROPOSITION XXIV. PROBLEM XVI. 

To describe a trajectory that shall pass through three given points, and 

touch two right lines given by position. 

Suppose HI, KL to be the given tangents 
and B, C, D, the given points. Through any 
two of those points, as B, D, draw the indefi 
nite right line BD meeting the tangents in 
the points H, K. Then likewise through 
any other two of these points, as C, D, draw 
the indefinite right line CD meeting the tan 
gents in the points I, L. Cut the lines drawn 
in R and S, so that HR may be to KR as 
the mean proportional between BH and HD is to the mean proportional 
between BK and KD ; and IS to LS as the mean pioportional between 
CI and ID is to the mean proportional between CL and LD. But you 
may cut, at pleasure, either within or between the points K and H, I and 
L, or without them ; then draw RS cutting the tangents in A and P, and 
A and P will be the points of contact. For if A and P are supposed to 
be the points of contact, situated anywhere else in the tangents, and through 
any of the points H, I, K, L, as I, situated in either tangent HI, a right 
line IY is drawn parallel to the other tangent KL, and meeting the curve 
in X and Y, and in that right line there be taken IZ equal to a mean pro 
portional between IX and IY, the rectangle XIY or IZ 2 , will (by the pro 
perties of the conic sections) be to LP 2 as the rectangle CID is to the rect 
angle CLD, that is (by the construction), as SI is to SL 2 ; and therefore 




SEC. V.] OF NATUKAL PHILOSOPHY. 141 

IZ is to LP as SI to SL. Wherefore the points S, P, Z. are in one right 
line. Moreover, since the tangents meet in G, the rectangle XI Y or IZ 2 
will (by the properties of the conic sections) be to IA 2 as GP 2 is to GA 2 , 
and consequently IZ will be to I A as GP to GA. Wherefore the points 
P, Z, A, lie in one right line, and therefore the points S, P, and A are in 
one right line. And the same argument will prove that the points R, P, 
and A are in one right line. Wherefore the points of contact A and P lie 
in the right line RS. But after these points are found, the trajectory may 
be described, as in the first Case of the preceding Problem. Q,.E.F. 

In this Proposition, and Case 2 of the foregoing, the constructions are 
the same, whether the right line XY cut the trajectory in X and Y, or 
not ; neither do they depend upon that section. But the constructions 
being demonstrated where that right line does cut the trajectory, the con 
structions where it does not are also known ; and therefore, for brevity s 
sake, I omit any farther demonstration of them. 

LEMMA XXII. 

To transform figures into other figures of the same kind. 

Suppose that any figure HGI is to be 
transformed. Draw, at pleasure, two par 
allel lines AO, BL, cutting any third line 
AB, given by position, in A and B, and from 
any point G of the figure, draw out any 
right line GD, parallel to OA, till it meet 
the right line AB. Then from any given 
point in the line OA, draw to the point 
D the right line OD, meeting BL in d ; and 
from the point of concourse raise the right 
line dg containing any given angle with the right line BL, and having 
such ratio to Qd as DG has to OD ; and g will be the point in the new 
figure hgi, corresponding to the point G. And in like manner the several 
points of the first figure will give as many correspondent points of the new 
figure. If we therefore conceive the point G to be carried along by a con 
tinual motion through all the points of the first figure, the point g will 
be likewise carried along by a continual motion through all the points of 
the new figure, and describe the same. For distinction s sake, let us call 
DG the first ordinate, dg the new ordinate, AD the first abscissa, ad the 
new abscissa ; O the pole. OD the abscinding radius, OA the first ordinate 
radius, and Oa (by which the parallelogram OABa is completed) the new 
ordinate radius. 

I say, then, that if the point G is placed in a right line given by posi 
tion, the point g will be also placed in a right line given by position. If 
the point G is placed in a conic section, the point g will be likewise placed 




J42 THE MATHEMATICAL PRINCIPLES [BOOK 1. 

in a conic section. And here I understand the circle as one of the conic 
sections. But farther, if the point G is placed in a line of the third ana 
lytical order, the point g will also be placed in a line of the third order, 
and so on in curve lines of higher orders. The two lines in which the 
points G, g, are placed, will be always of the same analytical order. For 
as ad is to OA, so are Od to OD, dg to DG, and AB to AD ; and there- 

OA X AB OA X dg 

fore AD is equal to , , and DG equal to 7 . Now if the 

ad ad 

point G is placed in a right line, and therefore, in any equation by which 
the relation between the abscissa AD and the ordinate GD is expressed, 
those indetermined lines AD and DG rise no higher than to one dimen- 

v v xu- ,. OA X AB . OA X dg 

sion, by writing this equation . m place of AD, and -. - 

in place of DG, a new equation will be produced, in which the new ab 
scissa ad and new ordinate dg rise only to one dimension ; and which 
therefore must denote a right line. But if AD and DG (or either of 
them) had risen to two dimensions in the first equation, ad and dg would 
likewise have risen to tAvo dimensions in the second equation. And so on 
in three or more dimensions. The indetermined lines, ad } dg in the 
second equation, and AD, DG, in the first, will always rise to the same 
number of dimensions ; and therefore the lines in which the points G, g, 
are placed are of the same analytical order. 

I say farther, that if any right line touches the curve line in the first 
figure, the same right line transferred the same way with the curve into 
the new figure will touch that curve line in the new figure, and vice versa. 
For if any two points of the curve in the first figure are supposed to ap 
proach one the other till they come to coincide, the same points transferred 
will approach one the other till they come to coincide in the new figure ; 
and therefore the right lines with which those points are joined will be 
come together tangents of the curves in both figures. I might have given 
demonstrations of these assertions in a more geometrical form ; but I study 
to be brief. 

Wherefore if one rectilinear figure is to be transformed into another, we 
need only transfer the intersections of the right lines of which the first 
figure consists, and through the transferred intersections to draw right lines 
in the new figure. But if a curvilinear figure is to be transformed, we 
must transfer the points, the tangents, and other right lines, by means of 
which the curve line is denned. This Lemma is of use in the solution of 
the more difficult Problems ; for thereby we maj transform the proposed 
figures, if they are intricate, into others that are more simple. Thus any 
right lines converging to a point are transformed into parallels, by taking 
for the first ordinate radius any right line that passes through the point 
of concourse of the converging lines, and that because their point of con- 



SEC. V.] OF NATURAL PHILOSOPHY. 143 

course is by this means made to go off in infinitum ; and parallel lines 
are such as tend to a point infinitely remote. And after the problem is 
solved in the new figure, if by the inverse operations we transform the 
new into the first figure, we shall have the solution required. 

This Lemma is also of use in the solution of solid problems. For as 
often as two conic sections occur, by the intersection of which a problem 
may be solved, any one of them may be transformed, if it is an hyperbola 
or a parabola, into an ellipsis, and then this ellipsis may be easily changed 
into a circle. So also a right line and a conic section, in the construc 
tion of plane problems, may be transformed into a right line and a circle 

PROPOSITION XXV. PROBLEM XVII. 

To describe a trajectory that shall pass through two given points, and 

touch three right lines given by position. 

Through the concourse of any two of the tangents one with the other, 
and the concourse of the third tangent with the right line which passes 
through the two given points, draw an indefinite right line ; and, taking 
this line for the first ordinate radius, transform the figure by the preceding 
Lemma into a new figure. In this figure those two tangents will become 
parallel to each other, and the third tangent will be parallel to the right 
line that passes through the two given points. Suppose hi, kl to be those 
two parallel tangents, ik the third tangent, and hi a right line parallel 
thereto, passing through those points a, b, 
through which the conic section ought to pass 
in this new figure; and completing the paral- 
lelogra n fiikl, let the right lines hi, ik, kl be 
BO cut in c, d, e, that he may be to the square 
root of the rectangle ahb, ic, to id, and ke to 
kd. as the sum of the right lines hi and kl is 
to the sum of the three lines, the first whereof 
is the right line ik, and the other two are the 

square roots of the rectangles ahb and alb ; and c, d, e, will be the points 
of contact. For by the properties of the conic sections, he 2 to the rectan 
gle ahb, and ic 2 to id 2 , and ke 2 to kd 2 , and el 2 to the rectangle alb, are all 
in the same ratio ; and therefore he to the square root of ahb, ic to id, ke 
to kdj and el to the square root of alb, are in the subduplicate of that 
ratio ; and by composition, in the given ratio of the sum of all the ante 
cedents hi + kl y to the sum of all the consequents ^/ahb -\- ik : *Jalb, 
Wherefore from that given ratio we have the points of contact c, d, e, in 
the new figure. By the inverted operations of the last Lemma, let those 
points be transferred into the first figure, and the trajectory will be there 
described by Prob. XIV. Q.E.F. But according as the points a, b, fall 
between the points //, /, or without taem, the points c, d, e, must be taken 



144 THE MATHEMATICAL PRINCIPLES BOOK I.J 

Cither between the points, h, i, k, /, or without them. If one of the points 
a, b, falls between the points h, i, and the other xvithout the points h, I, 
the Problem is impossible. 

PROPOSITION XXVI. PROBLEM XVIII. 

To describe a trajectory that shall pass through a given point, and touch 

four right lines given by position. 

From the common intersections, of any two 
of the tangents to the common intersection of 
the other two, draw an indefinite right line ; and 
taking this line for the first ordinate radius ; / x s o 
transform the figure (by Lem. XXII) into a new 
figure, and the two pairs of tangents, each of 
which before concurred in the first ordinate ra- 



dius, will now become parallel. Let hi and kl, Al l\ 

ik and hi, be those pairs of parallels completing the parallelogram hikl. 
And let p be the point in this new figure corresponding to the given point 
in the first figure. Through O the centre of the figure draw pq.: and O? 
being equal to Op, q will be the other point through which the conic sec 
tion must pass in this new figure. Let this point be transferred, by the 
inverse operation of Lem. XXII into the first figure, and there we shall 
have the two points through which the trajectory is to be described. But 
through those points that trajectory may be described by Prop. XVII. 



LEMMA XXIII. 

If two right lines, as AC, BD given by position, and terminating in 

given points A, B, are in a given ratio one to the other, and the right 

line CD, by which the, indetermined points C, D are joined is cut in 

K in a given ratio ; I say, that the point K will be placed in a right 

line given by position. 

For let the right lines AC, BD meet in 

E, and in BE take BG to AE as BD is to 

AC, and let FD be always equal to the given 

line EG ; and, by construction, EC will be 

to GD, that is, to EF, as AC to BD, and 

therefore in a given ratio ; and therefore the %- ,.--- I \ 

triangle EFC will be given in kind. Let E K cT^"^ 
CF be cut in L so as CL may be to CF in the ratio of CK to CD ; and 
because that is a given ratio, the triangle EFL will be given in kind, and 
therefore the point L will be placed in the right line EL given by position. 
Join LK, and the triangles CLK, CFD will be similar ; and because FD 
is a given line, and LK is to FD in a given ratio, LK will be also given 




SEC. V.] OF NATURAL PHILOSOPHY. 145 

To this let EH be taken equal, and ELKH will be always a parallelogram. 
And therefore the point K is always placed in the side HK (given by po 
tiition) of that parallelogram. Q.E.D. 

COR. Because the figure EFLC is given in kind, the three right lines 
EF, EL, and EC, that is, GD, HK, and EC, will have given ratios to 
each other. 

LEMMA XXIV. 

If three right lines, two whereof are parallel, and given by position, touch 
any conic section ; I say, that the semi-diameter of the section wkiJt 
is parallel to those two is a mean proportional between the segments 
of those two that are intercepted between the points of contact and the. 
third tangent. 

Let AF, GB be the two parallels touch 
ing the conic section ADB in A and B ; 
EF the third right line touching the conic 
section in I, and meeting the two former 
tangents in F and G, and let CD be the 
semi-diameter of the figure parallel to 
those tangents ; I say. that AF, CD, BG 
are continually proportional. 

For if the conjugate diameters AB, DM G Q 

meet the tangent FG in E and H, and cut one the other in C ; and the 
parallelogram IKCL be completed ; from the nature of the conic sections, 
EC will be to CA as CA to CL ; and so by division, EC CA to CA - 
CL, orEAto AL; and by composition, EA to EA + AL or EL, as EC to 
EC + CA or EB ; and therefore (because of the similitude of the triangles 
EAF, ELI, ECH, EBG) AF is to LI as CH to BG. Likewise, from tli? 
nature of the conic sections, LI (or CK) is to CD as CD to CH ; and 
therefore (ex aquo pertnrhatfy AF is to CD as CD to BG. Q.E.D. 

COR. 1. Hence if two tangents FG, PQ, meet two parallel tangents AF, 
BG in F and G, P and Q,, and cut one the other in O; AF (ex cequo per- 
tnrbot, ) will be to BQ as AP to BG, and by division, as FP to GQ, and 
therefore as FO to OG. 

COR. 2. Whence also the two right lines PG, FQ, drawn through the 
points P and G, F and Q, will meet in the right line ACB passing through 
the centre of the figure and the points of contact A, B. 

LEMMA XXV. 

If four sides of a parallelogram indefinitely produced touch any conic 
section, and are cut by a fifth tangent ; I say, that, taking those seg 
ments of any two conterminous sides that terminate in opposite angles 

10 




146 



THE MATHEMATICAL PRINCIPLES 



[BooK 1. 




of the parallelogram, either segment is to the side from which it is 

cut off as that part of the other conterminous side which is intercepted 

between the point of contact and the third side is to Uie other segment, 

Let the four sides ML, IK, KL, MI, 
of the parallelogram ML JK touch the 
conic section in A, B, C, I) ; and let the 
fifth tangent FQ cut those sides in F, 
Q, H, and E : and taking the segments 
ME, KQ of the sides Ml, KJ, or the 
segments KH, MF of the sides KL, 
ML, 1 s/.y, that ME is to MI as BK to 
KQ; and KH to KL as AM to MF. 
For, by Cor. 1 of the preceding Lemma, ME is to El as (AM or) BK to 
BQ ; and, by composition, ME is to MI as BK to KQ. Q.E.D. Also 
KH is to HL as (BK or) AM to AF ; and by division, KH to KL as AM 
to MF. Q.E.D. 

COR. 1. Hence if a parallelogram IKLM described about a given conic 
section is given, the rectangle KQ X ME, as also the rectangle KH X ME 
equal thereto, will be given. For, by reason of the similar triangles KQH 
MFE, those rectangles are equal. 

COR. 2. And if a sixth tangent eq is drawn meeting the tangents Kl. 
MI in q and e, the rectangle KQ X ME will be equal to the rectangle 
K</ X Me, and KQ will be to Me as Kq to ME, and by division ns 
Q? to Ee. 

COR. 3. Hence, also, if E<?, eQ, are joined and bisected, and a right line 
is drawn through the points of bisection, this right line will pass through 
the centre of the conic section. For since Q</ is to Ee as KQ to Me, the 
same right line will pass through the middle of all the lines Eq, eQ, MK 
(by Lem. XXIII), and the middle point of the right line MK is the 
centre of the section. 

PROPOSITION XXVII. PROBLEM XIX. 

To describe a trajectory that may touch jive right lines given by position. 

Supposing ABG ; BCF, 
GCD, FDE, EA to be the 

tangents given by position. 
Bisect in M and N, AF, BE, 
the diagonals of the quadri 
lateral 
tained 



figure 
under 



ABFE con- 
any four of 
them ; and (by Cor. 3, Lem. 
XXV) the right line MN 
draAvn through the points (,f 




SEC. V.] OF NATURAL PHILOSOPHY. 147 

bisection will pass through the centre of the trajectory. Again, bisect in 
P and Q, the diagonals (if I may so call them) Bl), GF of the quadrila 
teral figure EC OF contained under any other four tangents, and the right 
line PQ, drawn through the points of bisection will pass through the cen 
tre of the trajectory ; and therefore the centre will be given in the con 
course of the bisecting lines. Suppose it to be O. Parallel to any tan 
gent BG draw KL at such distance that the centre O may be placed in the 
middle between the parallels; this KL will touch the trajectory to be de 
scribed. Let this cut any other two tangents GCD, FJ)E, in L and K. 
Through the points G and K, F and L, where the tangents not parallel, 
CL, FK meet the parallel tangents CF, KL, draw GK, FL meeting in 
K ; and the right line OR drawn and produced, will cut the parallel tan 
gents GF, KL, in the points of contact. This appears from Gor. 2, Lem. 
XXIV. And by the same method the other points of contact may be 
found, and then the trajectory may be described by Prob. XIV. Q.E.F. 

SCPIOLTUM. 

Under the preceding Propositions are comprehended those Problems 
wherein either the centres or asymptotes of the trajectories are given. For 
when points and tangents and the centre are given, as many other points 
and as many other tangents are given at an equal distance on the other 
side of the centre. And an asymptote is to be considered as a tangent, ami 
its infinitely remote extremity (if we may say so) is a point of contact. 
Conceive the point of contact of any tangent removed in infinitum, and 
the tangent will degenerate into an asymptote, and the constructions of 
the preceding Problems will be changed into the constructions of those 
Problems wherein the asymptote is given. 

After the trajectory is described, we may 
find its axes and foci in this manmr. In the 
construction and figure of Lem. XXI, let those , 

legs BP, CP, of the moveable angles PEN, ^ 
PCN, by the concourse of which the trajec- \ 
tory was described, be made parallel one to 
the other : and retaining that position, let 
them revolve about their poles I , C, in that 
figure. In the mean while let the other legs 
GN, BN, of those angles, by their concourse 
K or k, describe the circle BKGC. Let O be the centre of this circle; 
and from this centre upon the ruler MN, wherein those legs CN, BN did 
concur while the trajectory was described, let fall the perpendicular OH 
meeting the circle in K and L. And when those other legs CK, BK meet 
in the point K that is nearest to the ruler, the first legs CP, BP will be 
parallel to the greater axis, and perpendicular on the lesser ; and the con- 




148 



THE MATHEMATICAL PRINCIPLES 



[Book I 




trary will hajpen if those legs meet in the remotest point L. Whence ii 
the centre of the trajectory is given, the axes will be given ; and those be- 
ing given, the foci will be readily found. 

But the squares of the axes are one to the 
other as KH to LH, and thence it is easy to 
describe a trajectory given in kind through 
f mr given points. For if two of the given 
points are made the poles C, 13, the third will 
give the moveable angles PCK, PBK ; but 
those being given, the circle BGKC may be 
described. Then, because the trajectory is 
given in kind, the ratio of OH to OK, and 
and therefore OH itself, will be given. About 
the centre O, with the interval OH, describe another circle, and the right 
line that touches this circle, and passes through the concourse of the legs 
CK, BK, when the first legs CP ; BP meet in the fourth given point, will 
be the ruler MN, by means of which the trajectory may be described 
Whence also on the other hand a trapezium given in kind (excepting a 
few cases that are impossible) may be inscribed in a given conic section. 

There are also other Lemmas, by the help of which trajectories given in 
kind may be described through given points, and touching given lines. 
Of such a sort is this, that if a right line is drawn through any point 
given by position, that may cut a given conic section in two points, and 
the distance of the intersections is bisected, the point of bisection will 
to ich ano her conic section of the same kind with the former, arid havin^ 
its axes parallel to the axes of the former. But I hasten to things of 
greater use. 



LEMMA XXVI. 

To place 1ht lit rev angles of a triangle, given both in kind and magni 
tude, in, respect of as many rigid lines given by position, -provided th\] 
are not all parallel among themselves, in such manner tfia t j ic spiral 
angles may touch the several lines. 

Three indefinite right lines AB, AC, BC, are 
given by position, and it is required so to place 
the triangle DEF that its angle 1) may touch 
the line AB, its angle E the line AC, and 
its angle F the line BC. Upon DE, DF, and 
EF, describe three segments of circles DRE, 
DGF. EMF, capable of angles equal to the 
Rubles BAG, ABC, ACB respectively. But those segments are to be de 
scribed t wards such sides of the lines DE, DF ; EF ; that the letters 




3 EC. V.I 



OF NATURAL PHILOSOPHY. 



1411 



DRED may turn round about in the same order with the letters I1ACB : 
the letters DGFD in the same order with the letters ABCA ; and the 
letters EMFE in the same order with the letters ACBA ; then, completing 
th se segmerts into entire circles let the two former circles cut one the 
other in G, and suppose P and Q to be their centres. Then joining GP, 
PQ, take Ga to AB as GP is to PQ ; and about the centre G, with the 
interval Ga, describe a circle that may cut the first circle DGE in a. 
Join aD cutting the second circle DFG in b, as well as aE cutting the 
third circle EMF in c. Complete the figure ABCdef similar and equal 
to the figure a&cDEF : I say, the thing is done. 

For drawing Fc meeting D in n, 
and joining aG ; bG, QG, QD. PD, by 
construction the angle EaD is equal to 
the angle CAB, and the angle acF equal 
to the angle ACB; and therefore the 
triangle aiic equiangular to the triangle 
ABC. Wherefore the angle anc or FnD 
is equal to the angle ABC, and conse- 
< uently to the angle F/>D ; and there 
fore the point n falls on the point b, 
Moreover the angle GPQ, which is half 
the angle GPD at the centre, is equal 
to the angle GaD at the circumference \ 
and the angle GQP, which is half the angle GQD at the centre, is equal 
to the complement to two right angles of the angle GbD at the circum 
ference, and therefore equal to the angle Gba. Upon which account the 
triangles GPQ, Gab, are similar, and Ga is to ab as GP to PQ. ; that is 
(by construction), as Ga to AB. Wherefore ab and AB are equal; and 
consequently the triangles abc, ABC, which we have now proved to be 
similar, are also equal. And therefore since the angles I), E, F, of the 
triangle DEF do respectively touch the sides ab, ar, be of the triangle 
afjc / the figure AECdef may be completed similar and equal to the figure 
afrcDEFj and by completing it the Problem will be solved. Q.E.F. 

COR. Hence a right line may be drawn whose parts given in length may 
be intercepted between three right lines given by position. Suppose the 
triangle DEF, by the access of its point D to the side EF, arid by having 
the sides DE, DF placed i>t directum to be changed into a right line 
whose given part DE is to be interposed between the right lines AB ; AC 
given by position; and its given part DF is to be interposed between the 
right lines AB ; BC, given by position; then, by applying the preceding 
construction to this case, the Problem will be solved. 




THE MATHEMATICAL PRINCIPLES 



[BOOK 1. 




PROPOSITION XXVIII. PROBLEM XX. 

To describe a trajectory giren both in kind and magnitude, given parts 
of which shall be interposed between three right lines given by position. 
Suppose a trajectory is to be described that 

may be similar and equal to the curve line DEF, 
-and may be cut by three right lines AB, AC, 

BC, given by position, into parts DE and EF, 

similar and equal to the given parts of this 

curve line. 

Draw the right lines DE, EF, DF: and 

place the angles D, E, F, of this triangle DEF, so 

as to touch those right lines given by position (by 

Lem. XXVI). Then about the triangle describe 

the trajectory, similar and equal to the curve DEF. 

Q.E.F. 

LEMMA XXVII. 

To describe a trapezium given in kind, the angles whereof may be , 
placed, in respect of four right lines given by position, that are neither 
all paralhl among themselves, nor converge to one common point, ////// 
the several angles may touch the several lines. 
Let the four right lines ABC, AD, BD, CE, be 
given by position ; the first cutting the second in A, 
the third in B, and the fourth in C and suppose a 
trapezium fghi is to be described that may be similar 
to the trapezium FCHI, and whose angle /, equal to 
the given angle F, may touch the right line ABC ; and 
(lie other angles g, h, i, equal to the other given angles, 
G, H, I, may touch the other lines AD, BD, CE, re 
spectively. Join FH, and upon FG. FH, FI describe J% 
as many segments of circles FSG, FTH, FVI, the first 
of which FSG may be capable of an angle equal to 
the angle BAD ; the second FTH capable of an angle 
equal to the angle CBD ; and the third FVI of an angle equal to the angle 
ACE. Bnrf>, the segments are to be described towards those sides of the 
lines FG, FH, FI, that the circular order of the letters FSGF may be 
the same as of the letters BADB, and that the letters FTHF may turn 
.ibout in the same order as the letters CBDC and the letters FVIF in the 
game order as the letters ACE A. Complete the segments into entire cir 
cles, and let P be the centre of the first circle FSG, Q, the centre of the 
second FTH. Join and produce both ways the line PQ,, and in it take 
QR in the same ratio to PQ as BC has to AB. But QR is to be taken 
towards that side of the point Q that the order of the letters P, Q,, R 




SEC. V.J 



OF NATURAL PHILOSOPHY. 



15] 




may be the same as of the letters A, B, C ; 
and about the centre R with the interval 
RF describe a fourth circle FNc cutting 
(lie third circle FVI in c. Join Fc 1 cut 
ting the first circle in a, and the second in 
/ . Draw aG, &H, cl, and let the figure 
ABC/ 4f/ii be made similar to the figure 
w^cFGHI; and the trapezium fghi will 
be that which was required to be de 
scribed. 

For let the two first circles FSG, FTH 
cut one the other in K ; join PK, Q,K, 
RK, "K, 6K, cK, and produce QP to L. 
The angles FaK, F6K, FcK at the circumferences are the halves of the 
angles FPK, FQK, FRK, at the centres, and therefore equal to LPK, 
LQ.K, LRK, the halves of those angles. Wherefore the figure PQRK is 
iquiangular and similar to the figure 6cK, and consequently ab is to be 
res PQ, to Q,R, that is, as AB to BC. But by construction, the angles 
Air, /B//,/C? , are equal to the angles FG, F&H, Fcl. And therefore 
the figure ABCfghi may be completed similar to the figure abcFGHl. 
vVliich done a trapezium fghi will be constructed similar to the trapezium 
FGHI, and which by its angles/, g, h, i will touch the right lines ABC, 
AD, BD, CE. Q.E.F. 

COR. Hence a right line may be drawn whose parts intercepted in a 
given order, between four right lines given by position, shall have a given 
proportion among themselves. Let the angles FGH, GHI, be so far in 
creased that the right lines FG, GH, HI, may lie in directum ; and by 
constructing the Problem in this case, a right line fghi will be drawn, 
whose parts fg, gh, hi, intercepted between the four right lines given by 
position, AB and AD, AD and BD, BD and CE, will be one to another 
as the lines FG, GH, HI, and will observe the same order among them 
selves. But the same thing may be more readily done in this manner. 

Produce AB to K and BD to L, 
so as BK may be to AB as HI to 
GH ; and DL to BD as GI to FG; 
and join KL meeting the right line 
CE in i. Produce iL to M, so as 
LM may be to iL as GH to HI ; 
then draw MQ, parallel to LB, and 
meeting the right line AD in g, and 
join gi cutting AB, BD in f, h ; I M* 
say, the thing is done. 

For let MO- cut the right line AB in Q, and AD the right line KL iu 





II 



^52 THE MATHEMATICAL PRINCIPLES [BOOK I. 

S, arid draw AP parallel to BD, and meeting iL in P, and -M to Lh (g\ 
to hi, Mi to Li, GI to HI, AK to BK) and AP to BL, will be in the same 
ratio. Cut DL in 11, so as DL to RL may be in that same ratio; and be 
cause ffS to g~M, AS to AP. and DS to DL are proportional; therefore 
(ex ceqit.o) as gS to LA, so will AS be to BL, and DS to RL ; and mixtly. 
BL RL to Lh BL, as AS DS to gS AS. That is, BR is to 
Eh as AD is to Ag, and therefore as BD to gQ. And alternately BR is 
to BD as 13/i to g-Q,, or as fh to fg. But by construction the line BL 
was cut in D and R in the same ratio as the line FI in G and H ; and 
therefore BR is to BD as FH to FG. Wherefore fh is to fg as FH to 
FG. Since, therefore, gi to hi likewise is as Mi to Li, that is, as GI to 
HI, it is manifest that the lines FI, fi, are similarly cut in G and H, g 
and //.. Q.E.F. 

In the construction of this Corollary, after the line LK is drawn cutting 
CE in i, we may produce iE to V, so as EV may be to Ei as FH to HI, 
arid then draw V/~ parallel to BD. It will come to the same, if about the 
centre i with an interval IH, we describe a circle cutting BD in X, and 
produce iX to Y so as iY may be equal to IF, and then draw Yf parallel 

to BO. 

Sir Christopher Wren and Dr. Wallis have long ago given other solu 
tions of this Problem. 

PROPOSITION XXIX. PROBLEM XXI. 

To describe a trajectory given in kind, that may be cut by four right 
lines given by position, into parts given in order, kind, and proportion. 
Suppose a trajectory is to be described that may be 
similar to the curve line FGHI, and whose parts, 
similar and proportional to the parts FG, GH, HI of 
the other, may be intercepted between the right lines 
AB and AD, AD, and BD, BD and CE given by po 
sition, viz., the first between the first pair of those lines, 
the second between the second, and the third between 
the third. Draw the right lines FG, GH, HI, FI; 
and (by Lem. XXVII) describe a trapezium fghi that 
may be similar to the trapezium FGHI, and whose an 
gles/, g, h, i, may touch the right lines given by posi 
tion AB, AD, BD, CE, severally according to their order. And then about 
bins trapezium describe a trajectory, that trajectory will be similar to the 
curve line FGHI. 

SCHOLIUM. 

This problem may be likewise constructed in the following manner. 
Joining FG, GH, HI, FI, produce GF to Y, and join FH, IG, and make 




SEC. VI 



OF NATURAL PHILOSOPHY. 



153 




El 



the angles CAK. DAL equal to 
the angles PGH, VFH. Let 

AK, AL meet the right line 
BD in K and L, and thence 
draw KM, LN, of which let 
KM make the angle A KM equal 
to the angle CHI, and be itself 
to AK as HI is to GH ; and let 

LN make the angle ALN equal to the angle FHI, and be itself 
to AL as HI to FH. But AK, KM. AL, LN are to be drawn 
towards those sides of the lines AD, AK, AL, that the letters 
OA.KMC, ALKA, DALND may be carried round in the same 
order as the letters FGHIF ; and draw MN meeting the right v 
line CE in L Make the angle iEP equal to the angle IGF, 
and let PE be to Ei as FG to GI ; and through P draw PQ/ that may 
with the right line ADE contain an angle PQE equal to the angle FIG, 
and may meet the right line AB in /, and join fi. But PE and PQ arc- 
to be drawn towards those sides of the lines CE, PE, that the circular 
order of the letters PEtP and PEQP may be the same as of the letters 
FGHIF ; and if upon the line/i, in the same order of letters, and similar 
to the trapezium FGHI, a trapezium /^//.i is constructed, and a trajectory 
given in kind is circumscribed about it, the Problem will be solved. 

So far concerning the finding of the orbits. It remains that we deter 
mine the motions of bodies in the orbits so found. 




SECTION VI. 

How the motions are to be found in given, orbits. 

PROPOSITION XXX. PROBLEM XXII. 

To find at any assigned time the place of a body moving in, a given 

parabolic trajectory. 

Let S be the focus, and A the principal vertex of 
the parabola; and suppose 4AS X M equal to the 
parabolic area to be cut off APS, which either was 
described by the radius SP, since the body s departure 
from the vertex, or is to be described thereby before 
its arrival there. Now the quantity of that area to 
be cut off is known from the time which is propor 
tional to it. Bisect AS in G, and erect the perpendicular GH equal to 
3M, and a circle described about th j centre H, with the interval HS, will 
cut the parabola in the place P required. For letting fall PO perpendic 
ular on the axis, and drawing PH, there will be AG 2 -f- GH 2 (=.= HP 2 -_ 

AO^TAGJ* + PO GH| 2 ) = AO 2 + PO 2 2CA > ?G!I f PO 



A G S 




154 THE MATHEMATICAL PRINCIPLES [BOOK I 

AG* + GH 2 . Whence 2GH X PO ( AO 2 + PO 2 2GAO) = AO J 

PO 2 
-f | PO 2 . For AO 2 write AO X ; then dividing all the terms by 



2PO ; and multiplying them by 2AS, we shall have ^GH X AS (= IAO 



the area APO SPO)| = to the area APS. But GH was 3M, and 

therefore ^GH X AS is 4AS X M. Wherefore the area cut off APS is 
equal to the area that was to be cut off 4AS X M. Q.E.D. 

Con. 1. Hence GH is to AS as the time in which the body described 
the arc AP to the time in which the body described the arc between the 
vertex A and the perpendicular erected from the focus S upon the axis. 

COR. 2. And supposing a circle ASP perpetually to pass through the 
moving body P, the xelocity of the point H is to the velocity which the 
body had in the vertex A as 3 to 8; and therefore in the same ratio is 
the line GH to the right line which the body, in the time of its moving 
from A to P, would describe with that velocity which it had in the ver 
tex A. 

COR. 3. Hence, also, on the other hand, the time may be found in which 
the body has described any assigned arc AP. Join AP, and on its middle 
point erect a perpendicular meeting the right line GH in H, 

LEMMA XXVIII. 

There is no oval figure whose area, cut off by right lines at pleasure, can, 
be universally found by means of equations of any number of finite 
terms and dimensions. 

Suppose that within the oval any point is given, about which as a pole 
a right line is perpetually revolving with an uniform motion, while in 
that right line a mov cable point going out from the pole moves always 
forward with a velocity proportional to the square of that right line with 
in the oval. By this motion that point will describe a spiral with infinite 
circumgyrations. Now if a portion of the area of the oval cut off by that 
right line could be found by a finite equation, the distance of the point 
from the pole, which is proportional to this area, might be found by the 
same equation, and therefore all the points of the spiral might be found 
by a finite equation also ; and therefore the intersection of a right line 
given in position with the spiral might also be found by a finite equation. 
But every right line infinitely produced cuts a spiral in an infinite num 
ber of points ; and the equation by which any one intersection of two lines 
is found at the same time exhibits all their intersections by as many roots, 
and therefore rises to as many dimensions as there are intersections. Be 
cause two circles mutually cut one another in two points, one of those in- 



8FC. Vl.J OF NATURAL PHILOSOPHY. 155 

terscctions is not to be found but by an equation of two dimensions, fo 
which the other intersection may be also found. Because there may b(- 
four intersections of two conic sections, any one of them is not to be found 
universally, but by an equation of four dimensions, by which they may bi> 
all found together. For if those intersections are severally sought, be 
cause the law and condition of all is the same, the calculus will be the 
same in every case, and therefore the conclusion always the same, which 
must therefore comprehend all those intersections at once within itself, and 
exhibit them all indifferently. Hence it is that the intersections of the 
conic se" f ions with the curves of the third order, because they may amount 
to six, (\,me out together by equations of six dimensions ; and the inter 
sections of two curves of the third order, because they may amount to nine, 
come out together by equations of nine dimensions. If this did not ne 
cessarily happen, we might reduce all solid to plane Problems, and those 
higher than solid to solid Problems. But here i speak of curves irreduci 
ble in power. For if the equation by which the curve is defined may bo 
reduced to a lower power, the curve will not be one single curve, but com 
posed of two, or more, whose intersections may be severally found by different 
calculusses. After the same manner the two intersections of right lines 
with the conic sections come out always by equations of two dimensions ; the 
three intersections of right lines with the irreducible curves of the third 
urder by equations of three dimensions ; the four intersections of right 
lines with the irreducible curves of the fourth order, by equations of four 
dimensions ; and so on in iitfinitum. Wherefore the innumerable inter 
sections of a right line with a spiral, since this is but one simple curve 
and not reducible to more curves, require equations infinite in r- .imber of 
dimensions and roots, by which they may be all exhibited together. For 
the law and calculus of all is the same. For if a perpendicular is let fall 
from the pole upon that intersecting right line, and that perpendicular 
together with the intersecting line revolves about the pole, the intersec 
tions of the spiral will mutually pass the one into the other ; and that 
which was first or nearest, after one revolution, will be the second ; after 
two, the third ; and so on : nor will the equation in the mean time be 
changed but as the magnitudes of those quantities are changed, by which 
the position of the intersecting line is determined. Wherefore since those 
quantities after every revolution return to their first magnitudes, the equa 
tion will return to its first form ; and consequently one and the same 
equation will exhibit all the intersections, and will therefore have an infi 
nite number of roots, by which they may be all exhibited. And therefore 
the intersection of a right line with a spiral cannot be universally found by 
any finite equation ; and of consequence there is no oval figure whose area, 
cut off by right lines at pleasure, can be universally exhibited by an^ 
such equation. 



1 56 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1 



By the same argument, if the interval of the pole and point by which 
the spiral is described is taken proportional to that part of the perimeter 
of the oval which is cut off, it may be proved that the length of the peri 
meter cannot be universally exhibited by any finite equation. But here I 
speak of ovals that are not touched by conjugate figures running out in 
infinitvm. 

COR. Hence the area of an ellipsis, described by a radius drawn from 
the focus to the moving body, is not to be found from the time given by a 
finite equation ; and therefore cannot be determined by the description ol 
curves geometrically rational. Those curves I call geometrically rational, 
all the points whereof may be determined by lengths that are definable 
by equations ; that is, by the complicated ratios of lengths. Other curves 
(such as spirals, quadratrixes, and cycloids) I call geometrically irrational. 
For the lengths which are or are not as number to number (according to 
the tenth Book of Elements) are arithmetically rational or irrational. 
And therefore I cut off an area of an ellipsis proportional to the time in 
which it is described by a curve geometrically irrational, in the following 
manner. 

PROPOSITION XXXI. PROBLEM XXIII. 

To find the place of a body moving in a given elliptic trajectory at any 

assigned time. 

Suppose A to be 
the principal vertex, 
S the focus, and O 
the centre of the 
ellipsis A PB ; and 
let P be the place of 
the body to be found. 
Produce OA to G so 
as OG may be to OA 
as OA to OS. Erect 
the perpendicular GH; and about the centre O, with the interval OG, de 
scribe the circle* GEF ; and on the ruler GH, as a base, suppose the wheel 
GEF to move forwards, revolving about its axis, and in the mean time by 
its point A describing the cycloid ALL Which done, take GK to the 
perimeter GEFG of the wheel, in the ratio of the time in which the body 
proceeding from A described the arc AP, to the time of a whole revolution 
in the ellipsis. Erect the perpendicular KL meeting the cycloid in L ; 
then LP drawn parallel to KG will meet the ellipsis in P, the required 
place of the body. 

For about the centre O with the interval OA describe the semi-circle 
AQB, and let LP, produced, if need be, meet the arc AQ, in Q, and join 




SEC. VI. 



OF NATURAL PHILOSOPHY. 



157 



SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the 
perpendicular Sll. The area APS is as the area AQS, that is, as tlie 
difference between the sector OQA and the triangle OQS, or as the difLi- 
ence of the rectangles *OQ, X AQ, and -J.OQ X SR, that is, because . >,_ 
is given, as the difference between the arc AQ and the right line Sll : ai.;l 
therefore (because of the equality of the given ratios SR to the sine of the 
arc AQ,, OS to OA, OA to OG, AQ to GF; and by division, AQ Sii 
to GF sine of the arc AQ) as GK, the difference between the arc C 1 
and tlie sine of the arc AQ. Q.E.D. 




SCHOLIUM. 

But since the description of this curve 
is difficult, a solution by approximation 
will be preferable. First, then, let there 
be found a certain angle B which may 
be to an angle of 57,29578 degrees, 
which an arc equal to the radius subtends, 
as SH, the distance of the foci, to AB, 
the diameter of the ellipsis. Secondly, a certain length L, which may be to 
the radius in the same ratio inversely. And these being found, the Problem 
may be solved by the following analysis. By any construction (or even 
by conjecture), suppose we know P the place of the body near its true 
place jo. Then letting fall on the axis of the ellipsis the ordinate PR 
from the proportion of the diameters of the ellipsis, the ordinate RQ of 
the circumscribed circle AQB will be given ; which ordinate is the sine of 
the angle AOQ, supposing AO to be the radius, and also cuts the ellipsis 
in P. It will .be sufficient if that angle is found by a rude calculus in 
numbers near the truth. Suppose we also know the angle proportional to 
the time, that is, which is to four right a iules as the time in which tlie 
body described the arc A/?, to the time of one revolution in the ellipsis. 
Let this angle be N. Then take an angle D, which may be to the angle 
B as the sine of the angle AOQ to the radius ; and an angle E which 
may be to the angle N AOQ -fD as the length L to the same length 
L diminished by the cosine of the angle AOQ, when that angle is less 
than a right angle, or increased thereby when greater. In the next 
place, take an angle F that may be to the angle B as the sine of the angle 
1OQ H- E to the radius, and an angle G, that may be to the angle N- 
AOQ E -f F as the length L to the same length L diminished by the 
cosine of the angle AOQ + E, when that angle is less than a right angle, 
or increased thereby when greater. For the third time take an angle H, 
that may be to the angle B as the sine of the angle AOQ f- E 4- G to the 
radius; and an angle I to the angle N AOQ E G -f- H, as the 



58 THE MATHEMATICAL PRINCIPLES jB(OK 1. 

length L is to the same length L diminished by the cosine of the angle 
AOQ -f- E + G, when that angle is less than a right angle, or increased 
thereby when greater. And so we may proceed in infinitum. Lastly, 
take the angle AOy equal to the angle AOQ -f- E 4- G + I -\- } &c. and 
from its cosine Or and the ordinatejor, which is to its sine qr as the lesser 
axis of the ellipsis to the greater, \\ e shall have p the correct place of the 
body. When the angle N AOQ, -f D happens to be negative, the 
sign -|- of the angle E must be every where changed into , and the sign 
into +. And the same thing is to be understood of the signs of the angles 
G and I, when the angles N AOQ E -f F, and N AOQ E 
G + H come out negative. But the infinite series AOQ -f- E -f- G -|- I +, 
&c. converges so very fast, that it will be scarcely ever needful to pro 
ceed beyond the second term E. And the calculus is founded upon 
this Theorem, that the area APS is as the difference between the arc 
AQ and the right line let fall from the focus S perpendicularly upon the 
radius OQ. 

And by a calculus not unlike, the Problem 
is solved in the hyperbola. Let its centre be 
O, its vertex A, its focus S, and asymptote 
OK ; and suppose the quantity of the area to 
be cut off is known, as being proportional to 
the time. Let that be A, and by conjecture 
suppose we know the position of a rij;ht i ne 
SP, that cuts off an area APS near the truth. 
Join OP, and from A and P to the asymptote T A S 

draw AI, PK parallel to the other asymptote ; and by the table of loga 
rithms the area AIKP will be given, and equal thereto the area OPA, 
which subducted from the triangle OPS, will leave the area cut off APS. 
And by applying 2 APS 2 A, or 2 A 2 A PS, the double difference of 
the area A that was to be cut off, and the area APS that is cut off, to the 
line SN that is let fall from the focus S, perpendicular upon the tangent 
TP, we shall have the length of the chord PQ. Which chord PQ is to 
be inscribed between A and P, if the area APS that is cut off be greater 
than the area A that was to be cut off, but towards the contrary side of the 
point P, if otherwise : and the point Q will be the place of the body more 
accurately. And by repeating the computation the place may be found 
perpetually to greater and greater accuracy. 

And by such computations we have a general 
analytical resolution of the Problem. But the par 
ticular calculus that follows is better fitted for as 
tronomical purposes. Supposing AO, OB, OD, to 
be the semi-axis of the ellipsis, and L its latus rec 
tum, and D the difference betwixt the lesser semi- 





SEC. VII.] OF NATURAL PHILOSOPHY. J 59 

axis OD, and -,L the half of the latus rectum : let an angle Y be found, whose 
sine may be to the radius as the rectangle under that difference J), and 
AO 4- OD the half sum of the axes to the square of the greater axis AB. 
Find also an angle Z, whose sine may be to the radius as the double rec 
tangle under the distance of the foci SH and that difference D to triple 
the square of half the greater semi-axis AO. Those angles being once 
found, the place of the body may be thus determined. Take the angle T 
proportional to the time in which the arc BP was described, or equal to 
what is called the mean motion ; and an angle V the first equation of thr 
mean motion to the angle Y, the greatest first equation, as the sine of 
double the angle T is to the radius ; and an angle X, the second equation, 
to the angle Z, the second greatest equation, as the cube of the sine of the 
angle T is to the cube of the radius. Then take the angle BHP the mean 
motion equated equal to T + X + V, the sum of the angles T, V. X, 
if the angle T is less than a right angle; or equal to T + X V, the 
difference of the same, if that angle T is greater than one and less than 
two right angles ; and if HP meets the ellipsis in P, draw SP, and it will 
cut off the area BSP nearly proportional to the time. 

This practice seems to be expeditious enough, because the angles V and 
X, taken in second minutes, if you please, being very small, it will be suf 
ficient to find two or three of their first figures. But it is likewise 
sufficiently accurate to answer to the theory of the planet s motions. 
For even in the orbit of Mars, where the greatest equation of the centre 
amounts to ten degrees, the error will scarcely exceed one second. But 
when the angle of the mean motion equated BHP is found, the angle oi 
the true motion BSP, and the distance SP, are readily had by the known 
methods. 

And so far concerning the motion of bodies in curve lines. But it mav 
also come to pass that a moving body shall ascend or descend in a right 
line : and I shall now go on to explain what belongs to such kind of 
motions. 



SECTION VII. 

Concerning the rectilinear ascent and descent of bodies, 

PROPOSITION XXXII. PROBLEM XXIV. 

Supposing that the centripetal force is reciprocally proportional to tht 
square of tlie distance of the places from the centre ; it is required 
to define the spaces which a body, falling directly, describes in given 
times. 
CASE 1. If the body does not fall perpendicularly, it will (by Cor. I 



160 



THE MATHEMATICAL PRINCIPLES 



[BOOK I 




Prop. XIII) describe some conic section whose focus is A 
placed in the centre of force. Suppose that conic sec 
tion to be A RPB and its focus S. And, first, if the 
figure be an ellipsis, upon the greater axis thereof AB 
describe the semi-circle ADB, and let the right line 
I) PC pass through the falling body, making right angles 
with the axis; and drawing DS, PS, the area ASD will c 
be proportional to the area ASP, and therefore also to 
the time. The axis AB still reaiaining the same, let the 
breadth of the ellipsis be perpetually diminished, and s 
the area ASD will always remain proportional to the 
time. Suppose that breadth to be diminished in, in fruit um ; and the orbit 
APB in that case coinciding with the axis AB, and the focus S with the 
extreme point of the axis B, the body will descend in the right line AC 1 . 
and the area ABD will become proportional to the time. Wherefore the 
space AC will be given which the body describes in a given time by its- 
perpendicular fall from the place A, if the area ABD is taken proportional 
to the time, and from the point D the right line DC is let fall perpendic 
ularly on the right line AB. Q,.E.I. 

CASE 2. If the figure RPB is an hyperbola, on the 
same principal diameter AB describe the rectangular 
hyperbola BED ; and because the areas CSP, CB/P, 
SPy 13, are severally to the sev eral areas CSD, CBED, 
SDEB, in the given ratio of the heights CP, CD, and 
the area SP/B is proportional to the time in which 
the body P will move through the arc P/B. the area 
SDEB will be also proportional to that time. Let 
the latus rectum of the hyperbola RPB be diminished 
in infitiitum, the latus transversum remaining the 
same; and the arc PB will come to coincide with the 
right line CB, and the focus S, with the vertex B, A- 
and the right line SD with the right line BD. And therefore the area 
BDEB will be proportional to the time in which the body C, by its per 
pendicular descent, describes the line CB. Q.E.I. 

CASE 3. And by the like argument, if the figure 
RPB is a parabola, and to the same principal ver 
tex B another parabola BED is described, that 
may always remain given while the former para 
bola in whose perimeter the body P moves, by 
having its latus rectum diminished and reduced 
to nothing, comes to coincide with the line CB, 
the parabolic segment BDEB will be proportional if 
to the time in which that body P or C will descend to the centre S or B 
Q.K.T 





fl.l 



OF NATURAL PHILOSOPHY. 



PROPOSITION XXXIII. THEOREM IX. 

The tilings above found being supposed. I say, thai the, velocity of a Jai 
ling body in any place C is to the velocity of a body, describing a 
circle about the centre B at the distance BC, in, the subduplicate ratio 
of AC, the distance of the body from the remoter vertex A of the circle 
or rectangular hyperbola, to iAB, the principal semi-diameter of the 




Let AB, the common dia 
meter of both figures RPB, 
DEB, be bisected in O; and 
draw the right line PT that 
may touch the figure RPB 
in P, and likewise cut that 
common diameter AB (pro 
duced, if need be) in T; and 
let SY be perpendicular to 
this line, and BQ to this di 
ameter, and suppose the latus 
rectum of the figure RPB to 
be L. Prom Cor. 9, Prop. 
XVI, it is manifest that the 
velocity of a body, moving 
in the line RPB about the 
centre S, in any place P, is 
to the velocity of a body describing a circle about the same centre, at the 
distance SP, in the subduplicate ratio of the rectangle L X SP to SY 2 
Por by the properties of the conic sections ACB is to CP 2 as 2AO to L. 

2CP 5 X AO 

and therefore rrrr; is equal to L. Therefore those, velocities an 




o-- 



ACB 

to each other in the subduplicate ratio of 



CP 3 X AO X SP 



ACB 



toSY~. More 



over, by the properties of the conic sections, CO is to BO as BO to TV.? 
and (by composition or division) as CB to BT. Whence (by division cs 
composition) BO or + CO will be to BO as CT to BT, that is, AC 



CP 2 X AO X SP 

ACB" 



is equal to 



will be to AO as CP to BQ; and therefore 

~AO X BC * ^ W su PP ose GV, tne breadth of the figure RPB, to 

be diminished in infinitum, so as the point P may come to coincide with 
the point C, and the point S with the point B. and the line SP with the 
line BC, and the line SY with the line BQ; and the velocity of the body 
now descending perpendicularly in the line CB will be to the velocity of 

11 



162 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1 



a body describing a circle about the centre B, at the distance BC, in thr 

BQ 2 X AC X SP 

subduplicate ratio of - -r-^ - ^ - to SY 2 , that is (neglecting the ra- 



X Jo 

tios of equality of SP to BC, and BQ, 2 to SY 2 ), in the subduplicate ratio 
of AC to AO, or iAB. Q.E.D. 

COR. 1 . When the points B and S come to coincide, TC will become to 
TS as AC to AO. 

COR. 2. A body revolving in any circle at a given distance from the 
centre, by its motion converted upwards, will ascend to double its distance 

from the centre. 



PROPOSITION XXXIV. THEOREM X. 

If the. figure BED is a parabola, I say, that the velocity of a falling 
body in any place C is equal to the velocity by which a body may 
uniformly describe a circle about the centre B at half the interval BC 
For (by Cor. 7, Prop. XVI) the velocity of a 

body describing a parabola RPB about the cen 

tre S, in any place P, is equal to the velocity of 

a body uniformly describing a circle about the c 

same centre S at half the interval SP. Let the 

breadth CP of the parabola be diminished in 

itifiiiitirni, so as the parabolic arc P/B may come 

to coincide with the right line CB, the centre S s 

with the vertex B, and the interval SP with the 

interval BC, and the proposition will be manifest. Q.E.D. 

PROPOSITION XXXV. THEOREM XL 

The same things supposed, I say, that the area of the figure DES, de 
scribed by the indefinite radius SD, is equal to the area which a body 
with a radius equal to h df the latus rectum of the figure DES, by 
uniformly revolving about the centre S, may describe in the same tijiw. 



1 



JD/ 






AJ 




SEC. ni: 



OF NATURAL PHILOSOPHY. 



For suppose a body C in the smallest moment of time describes in fal 
ling the infinitely little line Cc. while another body K, uniformly revolv 
ing about the centre S in the circle OK/r, describes the arc KA:. Erect the 
perpendiculars CD, cd, meeting the figure DES in D, d. Join SD, Sf/. 
SK. SA* ; and draw Del meeting the axis AS in T, and thereon let fall the 
perpendicular SY. 

CASE 1. If the figure DES is a circle, or a rectangular hyperbola, bisect 
its transverse diameter AS in O, and SO will be half the latus rectum. 
And because TC is to TD as Cc to Dd, and TD to TS as CD to S Y ; 
ex aquo TC will be to TS as CD X Cc to SY X Dd. But (by Cor. 1, 
Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescence 
of the points D, d, the ultimate ratios of the lines are taken. Wherefore 
AC is to AO or SK as CD X Cc to S Y X Vd, Farther, the velocity of 
the descending body in C IF, to the velocity of a body describing a circle 
about the centre S, at the interval SC, in the subduplicate ratio of AC to 
AO or SK (by Pi-op. XXXIII) ; and this velocity is to the velocity of a 
body describing the circle OKA: in the subduplicate ratio of SK to SC 
(by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity to the last, that is, 
the little line Cc to the arc K/r, in the subduplicate ratio of AC to SC, 
that is, in the ratio of AC to CD. Wherefore CD X Cc is equal to AC 
X KA*, and consequently AC to SK as AC X KA: to SY X IW. and 
thence SK X KA: equal to SY X Drf, and iSK X KA: equal to SY X DC/, 
that is, the area KSA* equal to the area SDrf. Therefore in every moment 
of time two equal particles, KSA" and SDrf, of areas are generated, which, 
if their magnitude is diminished, and their number increased in iiifinif t-w, 
obtain the ratio cf equality, and consequently (by Cor. Lem. IV), the whole 
areas together generated are always equal. Q..E.D. 

CASE 2. But if the figure DES is a 
parabola, we shall find, as above. CD X 
Cc to SY X Df/ as TC to TS, that is, 
as 2 to 1 ; and that therefore |CD X Cc 
is equal to i SY X Vd. But the veloc 
ity of the falling body in C is equal to 
the velocity w r ith which a circle may be 
uniformly described at the interval 4SC 
(by Prop" XXXIV). And this velocity 
to the velocity with which a circle may 
be described with the radius SK, that is, 
the little line Cc to the arc KA , is (by 
Cor. 6, Prop. IV) in the subduplicate ratio of SK to iSC ; that is, in the 
ratio of SK to *CD. Wherefore iSK X KA: is equal to 4CD X Cc, and 
therefore equal to SY X T)d ; that is, the area KSA* is equal to the area 
SIW, as above. Q.E.D. 




164 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1. 



PROPOSITION XXXVI. PROBLEM XXV. 

To determine the times of the descent of a body falling from 

place A. 

Upon the diameter AS, the distance of the body from the 
centre at the beginning, describe the semi-circle ADS, as 
likewise the semi-circle OKH equal thereto, about the centre 
S. From any place C of the body erect the ordinate CD. 
Join SD, and make the sector OSK equal to the area ASD. 
It is evident (by Prop. XXXV) that the body in falling will 
describe the space AC in the same time in which another body, 
uniformly revolving about the centre S, may describe the arc 
OK. Q.E.F. M 



a given 




PROPOSITION XXXVII. PROBLEM XXVI. 

To define the times of the ascent or descent of a body projected upwards 

or downwards from a given place. 

Suppose the body to go oif from the given place G, in the direction of 
the line GS, with any velocity. In the duplicate ratio of this velocity to 
the uniform velocity in a circle, with which the body may revolve about 




\ 



H 




D 




the centre S at the given interval SG, take GA to AS. If that ratio is 
the same as of the number 2 to 1, the point A is infinitely remote ; in 
which case a parabola is to be described with any latus rectum to the ver 
tex S, and axis SG ; as appears by Prop. XXXIV. But if that ratio is 
less or greater than the ratio of 2 to 1, in the former case a circle, in the 
latter a rectangular hyperbola, is to be described on the diameter SA; as 
appears by Prop. XXXIII. Then about the centre S, with an interval 
equal to half the latus rectum, describe the circle H/vK ; and at the place 
G of the ascending or descending body, and at any other place C, erect the 
perpendiculars GI, CD, meeting the conic section or circle in I and D. 
Then joining SI, SD, let the sectors HSK, HS& be made equal to the 
segments SEIS, SEDS. and (by Prop. XXXV) the body G will describe 



SEC. VII.] 



OF NATURAL PHILOSOPHY. 



165 




the space GO in the same time in which the body K may describe t*he arc 
Kk. Q.E.F. 

PROPOSITION XXXVIII. THEOREM XII. 

Supposing that the centripetal force is proportional to the altitude or 
distance of places from the centre, I say, that the times and velocities 
of falling bodies, and the spaces which they describe, are respectively 
proportional to the arcs, and the right and versed sines of the arcs. 
Suppose the body to fall from any place A in the A. 
right line AS ; and about the centre of force S, with 
the interval AS, describe the quadrant of a circle AE ; 
and let CD be the right sine of any arc AD ; and the 
body A will in the time AD in falling describe the 
space AC, and in the place C will acquire the ve 
locity CD. 

This is demonstrated the same way from Prop. X, as Prop. XXX11 was 
demonstrated from Prop. XI. 

COR. 1. Hence the times are equal in which one body falling from the 
place A arrives at the centre S, and another body revolving describes the 
quadrantal arc ADE. 

COR. 2. Wherefore all the times are equal in which bodies falling from 
whatsoever places arrive at the centre. For all the periodic times of re 
volving bodies are equal (by Cor. 3 ; Prop. IV). 

PROPOSITION XXXIX. PROBLEM XXVIT. 

Supposing a centripetal force of any kind, and granting the quadra- 
tnres of curvilinear figures ; it is required to find the velocity of a bod)/, 
ascending or descending in a right line, in the several places through 
which it passes ; as also the time in which it will arrive at any place : 
and vice versa. 
Suppose the body E to fall from any place 

A in the right line AD EC ; and from its place 

E imagine a perpendicular EG always erected 

proportional to the centripetal force in that 

place tending to the centre C ; and let BFG 

be a curve line, the locus of the point G. And D 

in the beginning of the motion suppose EG to 

coincide with the perpendicular AB ; and the 

velocity of the body in any place E will be as 

a right line whose square is equal to the cur 
vilinear area ABGE. Q.E.I. 
In EG take EM reciprocally proportional to 




E 



366 THE MATHEMATICAL PRINCIPLES [BOOK 1 

a right line whose square is equal to the area ABGE, and let VLM be a 
curve line wherein the point M is always placed, and to which the right 
line AB produced is an asymptote; and the time in which the body in 
falling- describes the line AE, will be as the curvilinear area ABTVME. 
Q.E.I. 

For in the right line AE let there be taken the very small line DE of 
a given length, and let DLF be the place of the line EMG, when the 
body was in D ; and if the centripetal force be such, that a right line, 
whose square is equal to the area ABGE ; is as the velocity of the descend 
ing body, the area itself will be as the square of that velocity ; that is, if 
for the velocities in D and E we write V and V + I, the area ABFD will 
be as VY, and the area ABGE as YY + 2VI -f II; and by division, the 

area DFGE as 2 VI -f LI, and therefore ^ will be as -- ^r 



that is. if we take the first ratios of those quantities when just nascent, the 

2YI 

length DF is as the quantity -|yrr an( i therefore also as half that quantity 

1 X Y 

But the time in which the body in falling describes the very 



line DE, is as that line directly and the velocity Y inversely ; and 
the force will be as the increment I of the velocity directly and the time 
inversely ; and therefore if we take the first ratios when those quantities 

I X V 

are just nascent, as -jy==r-. that is, as the length DF. Therefore a force 

proportional to DF or EG will cause the body to descend with a velocity 
that is as the right line whose square is equal to the area ABGE. Q.E.D. 

Moreover, since the time in which a very small line DE of a given 
length may be described is as the velocity inversely, and therefore also 
inversely as a right line whose square is equal to the area ABFD ; and 
since the line DL. and by consequence the nascent area DLME, will be as 
(he same right line inversely, the time will be as the area DLME, and 
the sum of all the times will be as the sum of all the areas : that is (by 
Cor. Lern. IV), the whole time in which the line AE is described will be 
as the whole area ATYME. Q.E.D. 

COR. 1. Let P be the place from whence a body ought to fall, so as 
that, when urged by any known uniform centripetal force (such as 
gravity is vulgarly supposed to be), it may acquire in the place D a 
velocity equal to the velocity which another body, falling by any force 
whatever, hath acquired in that place D. In the perpendicular DF let 
there be taken DR., which may be o DF as that uniform force to 
the other force in the place D. Complete the rectangle PDRQ,, and cut 
iff the area. ABFD equal to that rectangle. Then A will be the place 



SEC. VII. I 



OF NATURAL PHILOSOPHY. 



10; 




from whence the other body fell. For com 
pleting the rectangle DRSE, since the area 
ABFD is to the area DFGE as VV to 2VI, 
and therefore as 4V to I, that is, as half the 
whole velocity to the increment of the velocity 
of the body falling by the unequable force ; and 
in like manner the area PQRD to the area 
DRSE as half the whole velocity to the incre 
ment of the velocity of the body falling by the 
uniform force ; and since those increments (by 
reason of the equality of the nascent times) 
are as the generating forces, that is, as the or- 
dinates DF, DR, and consequently as the nascent areas DFGE, DRSE : 
therefore, ex aq-uo, the whole areas ABFD, PQRD will be to one another 
as the halves of the whole velocities ; and therefore, because the velocities 
are equal, they become equal also. 

COR. 2. Whence if any body be projected either upwards or downwards 
with a given velocity from any place D, and there be given the law of 
centripetal force acting on it, its velocity will be found in any other place, 
as e, by erecting the ordinate eg, and taking that velocity to the velocity 
in the place D as a right line whose square is equal to the rectangle 
PQRD, either increased by the curvilinear area DFge, if the place e is 
below the place D, or diminished by the same area DFg-e, if it be higher, 
is to the right line whose square is equal to the rectangle PQRD alone. 

COR. 3. The time is also known by erecting the ordinate em recipro 
cally proportional to the square root of PQRD -f- or T)Fge, and taking 
the time in which the body has described the line De to the time in which 
another body has fallen with an uniform force from P, and in falling ar 
rived at D in the proportion of the curvilinear area DLme to the rectan 
gle 2PD X DL. For the time in which a body falling with an uniform 
force hath described the line PD, is to the time in which the same body 
has described the line PE in the subduplicate ratio of PD to PE ; that is 
(the very small line DE being just nascent), in the ratio of PD to PD -f 
^DE ; or 2PD to 2PD -f- DE, and, by division, to the time in which the 
body hath described the small line DE, as 2PD to DE, and therefore as 
the rectangle 2PD X DL to the area DLME ; and the time in which 
both the bodies described the very small line DE is to the time in which 
the body moving unequably hath described the line De as the area DLME 
to the area DLme ; and, ex aquo, the first mentioned of these times is to 
the last as the rectangle 2PD X DL to the area DLrae. 



163 THE MATHEMATICAL PRINCIPLES [BoOK I 

SECTION VIII. 

Of the invention of orbits wherein bodies will revolve, being acted upon 
by any sort of centripetal force. 

PROPOSITION XL. THEOREM XIII. 

// a body, acted upon by any centripetal force, is any how moved, and 
another body ascends or descends in a right line, and their velocities 
be equal in amj one case of equal altitudes, t/ieir velocities will be also 
equal at all equal altitudes. 

Let a body descend from A through D and E, to the centre 
(j : and let another body move from V in the curve line VIK&. 
From the centre C, with any distances, describe the concentric 
circles DI, EK, meeting the right line AC in I) and E ; and 
the curve VIK in I and K. Draw 1C meeting KE in N, and 
on IK let fall the perpendicular NT and let the interval DE 
or IN between the circumferences of the circles be very small ; K / 
and imagine the bodies in D and I to have equal velocities. 
Then because the distances CD and CI are equal, the centri 
petal forces in D and I will be also equal. Let those forces be k) 
expressed by the equal lineoke DE and IN ; and let the force 
IN (by Cor. 2 of the Laws of Motion) be resolved into two 
others, NT and IT. r l hen the force NT acting in the direction 
line NT perpendicular to the path ITK of the body will not at all affect 
or change the velocity of the body in that path, but only draw it aside 
from a rectilinear course, and make it deflect perpetually from the tangent 
of the orbit, and proceed in the curvilinear path ITK/j. That whole 
force, therefore, will be spent in producing this effect: but the other force 
IT, acting in the direction of the course of the body, will be all employed 
in accelerating it, and in the least given time will produce an acceleration 
proportional to itself. Therefore the accelerations of the bodies in D and 
I, produced in equal times, are as the lines DE, IT (if we take the first 
ratios of the nascent lines DE, IN, IK, IT, NT) ; and in unequal times as 
those lines and the times conjunctly. But the times in which DE and IK 
are described, are, by reason of the equal velocities (in D and I) as the 
spaces described DE and IK, and therefore the accelerations in the course 
of the bodies through the lines DE and IK are as DE and IT, and DE 
and IK conjunctly ; that is, as the square of DE to the rectangle IT into 
IK. But the rectangle IT X IK is equal to the square of IN, that is, 
equal to the square of DE ; and therefore the accelerations generated in 
the passage of the bodies from D and I to E and K are equal. Therefore 
the velocities of the holies in E and K are also equal, and by the same 
reasoning they will always be found equal in any subsequent equal dis 
tances. Q..E.D. 




SEC. VI11.J OF NATURAL PHILOSOPHY. 169 

By the same reasoning, bodies of equal velocities and equal distances 
from the centre will be equally retarded in their ascent to equal distances. 
Q.E.D. 

COR. 1. Therefore if a body either oscillates by hanging to a string, or 
by any polished and perfectly smooth impediment is forced to move in a 
curve line ; and another body ascends or descends in a right line, and their 
velocities be equal at any one equal altitude, their velocities will be also 
equal at all other equal altitudes. For by the string of the pendulous 
body, or by the impediment of a vessel perfectly smooth, the same thing 
will be effected as by the transverse force NT. The body is neither 
accelerated nor retarded by it, but only is obliged to leave its rectilinear 
course. 

COR. 2. Suppose the quantity P to be the greatest distance from the 
centre to which a body can ascend, whether it be oscillating, or revolving 
in a trajectory, and so the same projected upwards from any point of a 
trajectory with the velocity it has in that point. Let the quantity A be 
the distance of the body from the centre in any other point of the orbit ; and 
let the centripetal force be always as the power A n , of the quantity A, the 
index of which power n 1 is any number n diminished by unity. Then 
the velocity in every altitude A will be as v/ P 11 A", and therefore will 
be given. For by Prop. XXXIX, the velocity of a body ascending and 
descending in a right line is in tha t very ratio. 

PROPOSITION XLI. PROBLEM XXVTII. 

Supposing a centripetal force of any kind, and granting the quadra 
tures of curvilinear figures, it is required to find as well the trajecto 
ries in which bodies will move, as the times of their motions in the 
trajectories found. 
Let any centripetal force tend to 
the centre C, and let it be required 
to find the trajectory VIKAr. Let R, 
there be given the circle VR, described 
from the centre C with any interval 
CV; and from the same centre de 
scribe any other circles ID, KE cut 
ting the trajectory in I and K, and 
the right line CV in D and E. Then 
draw the right line CNIX cutting the c 

circles KE, VR in N and X, and the right line CKY meeting the circle 
VJi in Y. Let the points I and K be indefinitely near ; and let the body 
go on from V through I and K to k ; and let the point A be the place 
from whence anothe body is to fall, so as in the place D to acquire a ve 
locity equal to the velocity of the first body in I. And things remaining 
as in Prop. XXXIX, the lineola IK, described in the least given time 




THE MATHEMATICAL PRINCIPLES [BOOK 1 

trill be as the velocity, and therefore as the right line whose square is 
equal to the area ABFD, and the triangle ICK proportional to the time 
will be given, and therefore KN will be reciprocally as the altitude 1C : 
that is (if there be given any quantity Q, and the altitude 1C be called 

A), as -T-. This quantity call Z, and suppose the magnitude of Q, to 

oe such that in some case v/ABFD may be to Z as IK to KN, and then 
in all cases V ABFD will be to Z as IK to KN, and ABFD to ZZ as 
IK 2 to KN 2 , and by division ABFD ZZ to ZZ as IN 2 to KN 2 , and there- 



fore V ABFD ZZ to Z, or as IN to KN; and therefore A x KN 

Q. x IN 

\vill be equal to . Therefore since YX X XC is to A X KN 

ZZ 



Q. X IN x CX 2 
as CX 2 , to AA, the rectangle XY X XC will be equal to- 



AAv/ABFD ZZ. 

Therefore in the perpendicular DF let there be taken continually I)//, IV 

Q ax ex 2 

equal to , =. respectively, and 

2 v/ ABFD ZZ 2AA V ABFD ZZ 

let the curve lines ab, ac, the foci of the points b and c, be described : and 
from the point V let the perpendicular Va be erected to the line AC, cut 
ting off the curvilinear areas VD&a, VDra, and let the ordi nates Es: ? 
E#, be erected also. Then because the rectangle D& X IN or DbzR is 
equal to half the rectangle A X KN, or to the triangle ICK ; and the 
rectangle DC X IN or Dc.rE is equal to half the rectangle YX X XC, or 
to the triangle XCY; that is, because the nascent particles I)6d3, ICK 
of the areas VD/>#, VIC are always equal; and the nascent particles 
Dc^-E, XCY of the areas VDca, VCX are always equal : therefore the 
generated area VD6a will be equal to the generated area VIC, and there 
fore proportional to the time; and the generated area VDco- is equal to 
the generated sector VCX. If, therefore, any time be given during which 
the body has been moving from V, there will be also given the area pro 
portional to it VD/>; and thence will be given the altitude of the body 
CD or CI ; and the area VDca, and the sector VCX equal there o, together 
with its angle VCL But the angb VCI, and the altitude CI being given, 
there is also given the place I, in which the body will be found at the end 
of that time. Q.E.I. 

COR. 1. Hence the greatest and least altitudes of the bodies, that is, the 
apsides of the trajectories, may be found very readily. For the apsides 
are those points in which a right line 1C drawn through the centre falls 
perpendicularly upon the trajectory VTK; which comes to pass when the 
right lines IK and NK become equal; that is, when the area ABFD ig 
C nl to ZZ. 



OF NATURAL PHILOSOPHY. 



171 




SEC. VI1LJ 

COR. 2. So also the angle KIN, in which the trajectory at any place 
cuts the line 1C. may be readily found by the given altitude 1C of the 
body : to wit, by making the sine of that angle to radius as KN to IK 
that is, as Z to the square root of the area ABFD. 

COR. 3. If to the centre C, and the 
principal vertex V, there be described a 
conic section VRS ; and from any point 
thereof, as R, there be drawn the tangent T 
RT meeting the axis CV indefinitely pro 
duced in the point T ; and then joining C 
CR there be drawn the right line CP, Q- 
equal to the abscissa CT, making an angle VCP proportional to the sector 
VCR ; and if a centripetal force, reciprocally proportional to the cubes 
of the distances of the places from the centre, tends to the centre C ; and 
from the place V there sets out a body with a just velocity in the direc 
tion of a line perpendicular to the right, line CV; that body will proceed 
in a trajectory VPQ,, which the point P will always touch ; and therefore 
if the conic section VI\ S be an hyberbola, the body will descend to the cen 
tre ; but if it be an ellipsis, it will ascend perpetually, and go farther and 
farther off in infinilum. And, on the contrary, if a body endued with any 
velocity goes off from the place V, and according as it begins either to de 
scend obliquely to the centre, or ascends obliquely from it, the figure VRS 
be either an hyperbola or an ellipsis, the trajectory may be found by increas 
ing or diminishing the angle VCP in a given ratio. And the centripetal 
force becoming centrifugal, the body will ascend obliquely in the trajectory 
VPQ, which is found by taking the angle VCP proportional to the elliptic 
sector VRC, and the length CP equal to the length CT, as before. All these 
things follow from the foregoing Proposition, by the quadrature of a certain 
ourve, the invention of which, as being easy enough, for brevity s sake I omit. 

PROPOSITION XLII. PROBLEM XXIX. 

The law of centripetal force being given, it is required to find the motion 
of a body setting out from a given place, with a given velocity, in the 
direction of a given right line. 
Suppose the same things as in 

Ihe three preceding propositions; 

and let the body go off from 

the place I in the direction of the 

little line, IK, with the same ve 
locity as another body, by falling 

with an uniform centripetal force 

from the place P, may acquire in 

I); and let this uniform force be 

to the force with which the body 




1.72 THE MATHEMATICAL PRINCIPLES [BOOK 1. 

is at first urged in I, as DR to DF. Let the body go on towards k; and 
about the centre C, with the interval Ck, describe the circle ke, meeting 
the right line PD in e, and let there be erected the lines eg, ev, ew, ordi- 
nately applied to the curves BF*, abv } acw. From the given rectangle 
PDRQ, and the given law of centripetal force, by which the first body is 
acted on, the curve line BF* is also given, by the construction of Prop. 
XXVII, and its Cor. 1. Then from the given angle CIK is given the 
proportion of the nascent lines 1K ; KN ; and thence, by the construction 
of Prob. XXVIII, there is given the quantity Q,, with the curve lines abv, 
acw ; and therefore, at the end of any time Dbve, there is given both 
the altitude of the body Ce or Ck, and the area Dcwe, with the sector 
equal to it XCy, the angle 1C A:, and the place k } in which the body will 
then be found. Q.E.I. 

We suppose in these Propositions the centripetal force to vary in its 
recess from the centre according to some law, which any one may imagine 
at pleasure; but at equal distances from the centre to be everywhere the 
Bame. 

I have hitherto considered the motions of bodies in immovable orbits. 
It remains now to add something concerning their motions in orbits which 
revolve round the centres of force. 



SECTION IX. 

Of the motion of bodies in moveable orbits ; and of the motion of the 

apsides. 

PROPOSITION XLIII. PROBLEM XXX. 

Ft is required to make a body move in a trajectory that revolves about 
the centre of force in the same manner as another body in the same 
trajectory at rest. 
In. the orbit VPK, given by position, let the body 

P revolve, proceeding from V towards K. From 

the centre C let there be continually drawn Cp, equal 

to CP, making the angle VC/? proportional to the 

angle VCP ; and the area which the line Cp describes 

will be to the area VCP, which the line CP describes 

at the same time, ns the velocity of the describing 

line Cp to the velocity of the describing line CP ; 

that is, as the angle VC/? to the angle VCP, therefore in a given ratio, 

and therefore proportional to the time. Since, then, the area described by 

the line Cp in an immovable plane is proportional to the time, it is manifest 

that a body, being acted upon by a just quantity of centripetal force may 




SEC. L\.] 



OF NATURAL PHILOSOPHY. 



173 



revolve with the point p in the curve line which the same point p, by the 
method just now explained, may be made to describe an immovable plane. 
Make the angle VC^ equal to the angle PC/?, and the line Cu equal to 
CV, and the figure uCp equal to the figure VCP ; and the body being al 
ways in the point p } will move in the perimeter of the revolving figure 
nCp, and will describe its (revolving) arc up in the same time the* the 
other body P describes the similar and equal arc VP in the quiescov.t fig 
ure YPK. Find, then, by Cor. 5, Prop. VI., the centripetal force by which 
the body may be made to revolve in the curve line which the pom* p de 
scribes in an immovable plane, and the Problem will be solved. O/E.K. 

PROPOSITION XLIV. THEOREM XIV. 

The difference of the forces, by which two bodies may be madi, to KMVG 
equally, one in a quiescent, the other in the same orbit revolving, i 1 in 
a triplicate ratio of their common altitudes inversely. 
Let the parts of the quiescent or 
bit VP, PK be similar and equal to 
the parts of the revolving orbit up, 
pk ; and let the distance of the points 
P and K be supposed of the utmost 
smallness Let fall a perpendicular 
kr from the point k to the right line 
pC, and produce it to m, so that mr 
may be to kr as the angle VC/? to the /2\- 
angle VCP. Because the altitudes 
of the bodies PC and pV, KG and 
kC } are always equal, it is manifest 
that the increments or decrements of 
the lines PC and pC are always 
equal ; and therefore if each of the 
several motions of the bodies in the places P and p be resolved into two 
(by Cor. 2 of the Laws of Motion), one of which is directed towards the 
centre, or according to the lines PC, pC, and the other, transverse to the 
former, hath a direction perpendicular to the lines PC and pC ; the mo 
tions towards the centre will be equal, and the transverse motion of the 
body p will be to the transverse motion of the body P as the angular mo 
tion of the line pC to the angular motion of the line PC ; that is, as the 
angle VC/? to the angle VCP. Therefore, at the same time that the bodv 
P, by both its motions, comes to the point K, the body p, having an equal 
motion towards the centre, will be equally moved from p towards C ; arid 
therefore that time being expired, it will be found somewhere in the 
line mkr, which, passing through the point k, is perpendicular to the line 
pC ; and by its transverse motion will acquire a distance from the line 




174 THE MATHEMATICAL PRINCIPLES [BOOK J. 

C, that will be to the distance which the other body P acquires from the 
line PC as the transverse motion of the body p to the transverse motion of 
the other body P. Therefore since kr is equal to the distance which the 
body P acquires from the line PC, and mr is to kr as the angle VC/? to 
the angle VCP, that is, as the transverse motion of the body p to the 
transverse motion of the body P, it is manifest that the body p, at the ex 
piration of that time, will be found in the place m. These things will be 
so, if the bodies jo and P are equally moved in the directions of the lines 
pC and PC, and are therefore urged with equal forces in those directions. 
I: ut if we take an angle pCn that is to the angle pCk as the angle VGj0 
to the angle VCP, and nC be equal to kG, in that case the body p at the 
expiration of the time will really be in n ; and is therefore urged with a 
greater force than the body P, if the angle nCp is greater than the angle 
kCp, that is, if the orbit npk, move either in cmiseqnentia, or in antece- 
denticij with a celerity greater than the double of that with which the line 
CP moves in conseqnentia ; and with a less force if the orbit moves slower 
in antecedent-la. And ihj difference of the forces will be as the interval 
mn of the places through which the body would be carried by the action of 
that difference in that given space of time. About the centre C with the 
interval Cn or Ck suppose a circle described cutting the lines mr, tun pro 
duced in s and , and the rectangle mn X nit will be equal to the rectan- 

*//? n ^* */?? ^ 

"le mk X ins, and therefore mn will be equal to . But since 

mt 

the triangles pCk, pCn, in a given time, are of a given magnitude, kr and 
mr. a id their difference mk, and their sum ms, are reciprocally as the al 
titude pC, and therefore the rectangle mk X ms is reciprocally as the 
square of the altitude pC. But, moreover, mt is directly as |//z/, that is, as 
the altitude pC. These are the first ratios of the nascent lines ; and hence 

r - that is, the nascent lineola mn. and the difference of the forces 

mt 

proportional thereto, are reciprocally as the cube of the altitude pC. 
Q.E.D. 

COR. I. Hence the difference of the forces in the places P and p, or K and 
/.*, is to the force with which a body may revolve with a circular motion 
from R to K, in the same time that the body P in an immovable orb de 
scribes the arc PK, as the nascent line m,n to the versed sine of the nascent 

mk X ms rk 2 

arc RK, that is, as to ^g, or as mk X ms to the square of 

rk ; that is. if we take given quantities F and G in the same ratio to one 
another as the angle VCP bears to the angle VQ?, as GG FF to FF. 
And, therefore, if from the centre C, with any distance CP or Cp, there be 
described a circular sector equal to the whole area VPC, which the body 



OEC. 



IX.l 



OF NATURAL PHILOSOPHY. 



175 



revolving in an immovable orbit has by a radius drawn to the centre de- 
bribed in any certain time, the difference of the forces, with which the 
body P revolves in an immovable orbit, and the body p in a movable or 
bit, will be to the centripetal force, with which another body by a radius 
drawn to the centre can uniformly describe that sector in the same time 
as the area VPC is described, as GG FF to FF. For that sector and 
the area pCk are to one another as the times in which they are described. 

COR. 2. If the orbit YPK be an 
ellipsis, having its focus C, and its 
highest apsis Y, and we suppose the 
the ellipsis upk similar and equal to .. 
it, so that pC may be always equal / 
to PC, and the angle YC/? be to the ; 
angle YCP in the given ratio of G \ 
to F ; and for the altitude PC or pC \ 
we put A, and 2R for the latus rec- /t\ 
turn of the ellipsis, the force with * 
which a body may be made to re 
volve in a movable ellipsis will be as 

FF RGG RFF 

- + - -rg , and vice versa. 
/Y A. A. 

Let the force with which a body may 

revolve in an immovable ellipsis be expressed by the quantity , and the 




-. 7 



force in V will be 



FF 



But the force with which a body may revolve in 



a circle at the distance CY, with the same velocity as a body revolving in 
an ellipsis has in Y, is to the force with which a body revolving in an ellip 
sis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to the 

RFF 

semi-diameter CY of the circle, and therefore is as , =- : and tlu 



RFF 

which is to this, as GG FF to FF, is as - ~py^~~ ~ : and this force 

(by Cor. 1 cf this Prop.) is the difference of the forces in Y, with which the 
body P revolves in the immovable ellipsis YPK, and the body p in the 
movable ellipsis upk. Therefore since by this Prop, that difference at 

any other altitude A is to itself at the altitude CY as -r-, to ^TF- the same 

A J CY J 

R C^ ( "* R P^ T* 1 

difference in every altitude A will be as - 3 : . Therefore to the 

FF 
force -T-: , by which the body may revolve in an immovable ellipsis VPK 



176 THE MATHEMATICAL PRINCIPLES [BOOK I. 

idd the excess -:-= , and the sum will be the whole force -r-r -\- 

A AA 

RGG RFF, 

.-5 by which a body may revolve in the same time in the mot- 

A. 

able ellipsis upk. 

COR. 3. In the same manner it will be found, that, if the immovable or 
bit VPK be an ellipsis having its centre in the centre of the forces C } and 
there be supposed a movable ellipsis -upk, similar, equal, and concentrical 
to it ; and 2R be the principal latus rectum of that ellipsis, and 2T the 
latus transversum, or greater axis ; and the angle VCjo be continually to the 
angle TCP as G to F ; the forces with which bodies may revolve in the im- 

FFA FFA 

movable and movable ellipsis, in equal times, will be as ^ and -p~ 

RGG RFF 

+ .-3 respectively. 

A 

COR. 4. And universally, if the greatest altitude CV of the body be called 
T, and the radius of the curvature which the orbit VPK has in Y, that is, 
the radius of a circle equally curve, be called R, and the centripetal force 
with which a body may revolve in any immovable trajectory VPK at the place 

VFF 
V be called - f -=Trri , and in other places P be indefinitely styled X ; and the 

altitude CP be called A, and G be taken to F in the given ratio of the 
angle VCjD to the angle VCP ; the centripetal force with which the same 
body will perform the same motions in the same time, in the same trajectory 
upk revolving with a circular motion, will be as the sum of the forces X -f- 
VRGG VRFF 

~ A* 

COR. 5. Therefore the motion of a body in an immovable orbit being 
given, its angular motion round the centre of the forces may be increased 
or diminished in a given ratio; and thence new immovable orbits may be 
found in which bodies may revolve with new centripetal forces. 

COR. 6. Therefore if there be erected the line VP of an indeterminate 
-p length, perpendicular to the line CV given by po 

sition, and CP be drawn, and Cp equal to it, mak 
ing the angle VC/? having a given ratio to the an 
gle VCP, the force with which a body may revolve 
in the curve line Vjo/r, which the point p is con 
tinually describing, will be reciprocally as the cube 
C of the altitude Cp. For the body P, by its vis in 
ertia alone, no other force impelling it, will proceed uniformly in the right 
line VP. Add, then, a force tending to the centre C reciprocally as the 
cube of the altitude CP or Cp, and (by what was just demonstrated) the 




SEC. IX..J OF NATURAL PHILOSOPHY. 177 

body will deflect from the rectilinear motion into the curve line Ypk. But 
this curve ~Vpk is the same with the curve VPQ found in Cor. 3, Prop 
XLI, in which, I said, hodies attracted with such forces would ascend 
obliquely. 

PROPOSITION XLV. PROBLEM XXXL 

To find the motion of the apsides in orbits approaching very near to 

circles. 

This problem is solved arithmetically by reducing the orbit, which a 
body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above 
Prop.) describes in an immovable plane, to the figure of the orbit whose 
apsides are required ; and then seeking the apsides of the orbit which that 
body describes in an immovable plane. But orbits acquire the same figure, 
if the centripetal forces with which they are described, compared between 
themselves, are made proportional at equal altitudes. Let the point V be 
the highest apsis, and write T for the greatest altitude CV, A for any other 
altitude CP or C/?, and X for the difference of the altitudes C V CP : 
and the force w r ith which a body moves in an ellipsis revolving about its 

p -p T? C* f ^ T? F* F 

focus C (as in Cor. 2), and which in Cor. 2 was as -r-r -\ -.-3 , 

FFA + RGG RFF , 

that is as, -^ , by substituting T X for A, will be- 

A 

RGG RFF + TFF FFX 

come as -p . In like manner any other cen 
tripetal force is to be reduced to a fraction whose denominator is A 3 , and 
the numerators are to be made analogous by collating together the homo 
logous terms. This will be made plainer by Examples. 

EXAMPLE 1. Let us suppose the centripetal force to be uniform, 

A 3 
and therefore as 3 or, writing T X for A in the numerator, as 

T 3 3TTX + 3TXX X 3 

=-. Ihen collating together the correspon- 

A 3 

dent terms of the numerators, that is, those that consist of given quantities, 
with those of given quantities, and those of quantities not given with those 
of quantities not given, it will become RGG RFF -f- TFF to T 3 as 
FFX to 3TTX -f 3TXX X 3 , or as FF to 3TT + 3TX XX. 
Now since the orbit is supposed extremely near to a circle, let it coincide 
with a circle ; and because in that case R and T become equal, and X is 
infinitely diminished, the last ratios will be, as RGG to T 2 , so FF to 
3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TT 
to 3TT, that is, as 1 to 3 ; and therefore G is to F, that is, the angle VC/? 
to the angle VCP, as 1 to v/3. Therefore since the body, in an immovable 



178 THE MATHEMATICAL PRINCIPLES [BOOK I 

ellipsis, in descending from the upper to the lower apsis, describes an angle, 
if I may so speak, of ISO deg., the other body in a movable ellipsis, and there 
fore in the immovable orbit we are treating of, will in its descent from 

180 

the upper to the lower apsis, describe an angle VCjt? of ^ deg. And this 

\/o 

comes to pass by reason of the likeness of this orbit which a body acted 
upon by an uniform centripetal force describes, and of that orbit which a 
body performing its circuits in a revolving ellipsis will describe in a quies 
cent plane. By this collation of the terms, these orbits are made similar ; 
not universally, indeed, but then only when they approach very near to a 
circular figure. A body, therefore revolving with an uniform centripetal 

180 

force in an orbit nearly circular, will always describe an angle of deg/, or 

v/o 

103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to the 
lower apsis when it has once described that angle, and thence returning to 
the upper apsis when it has described that angle again ; and so on in in- 
finitwn. 

EXAM. 2. Suppose the centripetal force to be as any power of the alti- 

A n 

tude A, as, for example, A n 3 , or-r^ ; where n 3 and n signify any in- 

A. 

dices of powers whatever, whether integers or fractions, rational or surd, 
affirmative or negative. That numerator A n or T X| n being reduced to 
an indeterminate series by my method of converging series, will become 

T n >/XT n T + ^ XXT n 2 , &c. And conferring these terms 
with the terms of the other numerator RGG RFF + TFF FFX, it 
becomes as RGG RFF 4- TFF to T n , so FF to ?/.T n r + ? ~^ 

XT n 2 , &c. And taking the last ratios where the orbits approach to 
circles, it becomes as RGG to T 1 , so FF to nT- 1 T , or as GG to 
T" , so FF to ?*T n ; and again, GG to FF, so T n l to nT" 1 , that 
is, as 1 to n ; and therefore G is to F, that is the angle VCp to the angle 
VCP, as 1 to ^/n. Therefore since the angle VCP, described in the de 
scent of the body from the upper apsis to the lower apsis in an ellipsis, is 
of 180 deg., the angle VC/?, described in the descent of the body from the 
upper apsis to the lower apsis in an orbit nearly circular which a body de 
scribes with a centripetal force proportional to the power A n 3 , will be equal 

ISO 

to an angle of - deg., and this angle being repeated, the body will re- 
\/ti 

turn from the lower to the upper apsis, and so on in infinitum. As if the 
centripetal force be as the distance of the body from the centre, that is, as A, 

A 4 
or -p, n will be equal to 4, and ^/n equal to 2 ; and thereLre the angle 



IX.] OF NATURAL PHILOSOPHY. IT9 

ISO 

between the upper and the lower apsis will be equal to deg., or 90 deg. 

Therefore the body having performed a fourth part of one revolution, will 
arrive at the lower apsis, and having performed another fourth part, will 
arrive at the upper apsis, and so on by turns in infiuitum. This appears 
also from Prop. X. For a body acted on by this centripetal force will re 
volve in an immovable ellipsis, whose centre is the centre of force. If the 

1 A 2 

centripetal force is reciprocally as the distance, that is, directly as or 

A A" 

ji will be equal to 2 ; and therefore the angle between the upper and lower 

180 

apsis will be - deg., or 127 deg., 16 min., 45 sec. ; and therefore a body re 
v/2 

volving with such a force, will by a perpetual repetition of this angle, move 
alternately from the upper to the lower and from the lower to the upper 
apsis for ever. So. also, if the centripetal force be reciprocally as the 
biquadrate root of the eleventh power of the altitude, that is, reciprocally 

as A , and, therefore, directly as -r-fp or as Ts> n wil * ^e et l ual f \> an(1 
4 A^- A 

1 Of) 

- deg. will be equal to 360 deg. ; and therefore the body parting from 
v/ n 

the upper apsis, and from thence perpetually descending, will arrive at the 
lower apsis when it has completed one entire revolution ; and thence as 
cending perpetually, when it has completed another entire revolution, it 
will arrive again at the upper apsis ; and so alternately for ever. 

EXAM. 3. Taking m and n for any indices of the powers of the alti 
tude, and b and c for any given numbers, suppose the centripetal force 

6A ra + cA" b into T X> -f- c into T X 

to be as r^ that is, as 



A 3 A 3 

or (by the method of converging series above-mentioned) as 

bT m + cT n m6XT" - 1 //cXT n mm m vvrpm un n 

~~2 -- 0A.A1 ^ 



t-XXT" 2 , fcc. 

T$~ ~ and comparing the terms of the numerators, there will 

arise RGG II FF -f TFF to ^T m + cT" as FF to mbT m i 



" - + 2 " m bXT" - * + "^p cXTn - .fee. And tak- 

ing the last ratios that arise when the orbits come to a circular form, there 
will come forth GG to 6T m l -f cT n 1 as FF to mbT m l + ncT" J ; 
and again, GG to FF as 6T m + cT n to mbT n 1 -f ncT n \ 
This proportion, by expressing the greatest altitude CV or T arithmeti 
cally by unity, becomes, GG to FF as b -{- c to mb -\- ?/c, and therefore as I 



(80 THE MATHEMATICAL PRINCIPLES [BOOK 1 

tub ~h nc 
to - y7 Whence G becomes to P, that is, the angle VCjo to the an- 

f) ~T~ C 

gle VCP. as 1 to >/- . - -. And therefore since the angle VCP between 

the upper and the lower apsis, in an immovable ellipsis, is of 180 deg., thr 
angle VC/? between the same apsides in an orbit which a body describes 

b A m I c A n 

with a centripetal force, that is. as - r , will be equal to an angle of 

A. 

ISO v/ 1~TT~ ; deg. And y tne same reasoning, if the centripetal force 
be as - 73 , the angle between the apsides will be found equal to 

fi f* 

18o V - - deg. After the same manner the Problem is solved in 
nib >ic 

more difficult cases. The quantity to which the centripetal force is pro 
portional must always be resolved into a converging series whose denomi 
nator is A*. Then the given part of the numerator arising from that 
operation is to be supposed in the same ratio to that part of it which is not 
given, as the given part of this numerator RGG RFF -f TFF FFX. 
is to that part of the same numerator which is not given. And taking 
away the superfluous quantities, and writing unity for T, the proportion 
of G to F is obtained. 

COR. 1 . Hence if the centripetal force be as any power of the altitude, 
that power may be found from the motion of the apsides ; and so contra 
riwise. That is, if the whole angular motion, with which the body returns 
to the same apsis, be to the angular motion of one revolution, or 360 deg., 
MS any number as m to another as n, and the altitude called A ; the force 

nn 

will be as the power A HSii 3 of the altitude A; the index of which power is 
- 3. This appears by the second example. Hence it is plain that 

the force in its recess from the centre cannot decrease in a greater than a 
triplicate ratio of the altitude. A body revolving with such a force, and 
parting from the apsis, if it once begins to descend, can never arrive at the 
lower apsis or least altitude, but will descend to the centre, describing the 
curve line treated of in Cor. 3, Prop. XLL But if it should, at its part 
ing from the lower apsis, begin to ascend never so little, it will ascend in 
irtfimtifm, and never come to the upper apsis ; but will describe the curve 
line spoken of in the same Cor., and Cor. 6, Prop. XLIV. So that where 
the force in its recess from the centre decreases in a greater than a tripli 
cate ratio of the altitude, the body at its parting from the apsis, will either 
descend to the centre, or ascend in iiiftnitum, according as it descends or 
Ascends at the beginning of its motion. But if the force in its recess from 



"SEC. IX.J OF NATURAL PHILOSOPHY. ISi 

the centre either decreases in a less than a triplicate ratio of the altitude, 
or increases in any ratio of the altitude whatsoever, the body will never 
descend to the centre, but will at some time arrive at the lower apsis ; and, 
on the contrary, if the body alternately ascending and descending from one 
apsis to another never comes to the centre, then either the force increases 
in the recess from the centre, or it decreases in a less than a triplicate ratio 
of the altitude; and the sooner the body returns from one apsis to another, 
the farther is the ratio of the forces from the triplicate ratio. As if the 
body should return to and from the upper apsis by an alternate descent and 
ascent in 8 revolutions, or in 4, or 2, or \\ that is, if m should be to n as 8, 

or 4, or 2, or H to 1. and therefore --- 3, be g\ 3,or T V~ 3, or i 3, or 

mm 

3 



I - 3 ; then the force will be as A~ ? or A T "~ 3j or A*~~ 3j or A"" G 

that is. it will be reciprocally as A 3 C4 or A 3 T ^ or A 3 4 or A 3 "" 
If the body after each revolution returns to the same apsis, and the apsis 

nn _ 

remains unmoved, then m will be to n as 1 to 1, and therefore A" 

will be equal to A 2 , or - ; and therefore the decrease of the forces will 

A A 

be in a duplicate ratio of the altitude ; as was demonstrated above. If the 
body in three fourth parts, or two thirds, or one third, or one fourth part 
of an entire revolution, return to the same apsis ; m will be to n as or ? 

n n i_6 _ 3 9 _ 3 o 

or ^ or i to 1, and therefore Amm 3 is equal to A 9 or A 4 or A 

_ 3 1 6 _ 3 l_l 

or A ; and therefore the force is either reciprocally as A fl or 

3 613 

A 4 or directly as A or A . Lastly if the body in its progress from the 
upper apsis to the same upper apsis again, goes over one entire revolution 
and three deg. more, and therefore that apsis in each revolution of the body 
moves three deg. in consequentia ; then m will be to u as 363 deg. to 



360 deg. or as 121 to 120, and therefore Amm will be equal to 

2 9_ 5_ 2_ JJ 

A " and therefore the centripetal force will be reciprocally as 

^T4"6TT> or rec ip r ocally as A 2 ^ 4 ^ very nearly. Therefore the centripetal 
force decreases in a ratio something greater than the duplicate ; but ap 
proaching 59f times nearer to the duplicate than the triplicate. 

COR. 2. Hence also if a body, urged by a centripetal force which is re 
ciprocally as the square of the altitude, revolves in an ellipsis whose focus 
is in the centre of the forces ; and a new and foreign force should be added 
to or subducted from this centripetal force, the motion of the apsides arising 
from that foreign force may (by the third Example) be known ; and so on 
the contrary. As if the force with which the body revolves in the ellipsis 



182 THE MATHEMATICAL PRINCIPLES [BOOK I 

oe as -r-r- ; and the foreign force subducted as cA, and therefore the remain- 

A .A. 

^ c ^4 

ing force as -^ ; then (by the third Example) b will be equal to 1. 

m equal to 1, and n equal to 4 ; and therefore the angle of revolution be 

1 c 

tween the apsides is equal to 180 <*/- deg. Suppose that foreign force 

to be 357.45 parts less than the other force with which the body revolves 
in the ellipsis : that is, c to be -3 }y j ; A or T being equal to 1 ; and then 

l8( Vl~4 c will be 18<Vfff Jf or 180.7623, that is, 180 deg., 45 min., 

44 sec. Therefore the body, parting from the upper apsis, will arrive at 
the lower apsis with an angular motion of 180 deg., 45 min., 44 sec , and 
this angular motion being repeated, will return to the upper apsis ; and 
therefore the upper apsis in each revolution will go forward 1 deg., 31 min., 
28 sec. The apsis of the moon is about twice as swift 

So much for the motion of bodies in orbits whose planes pass through 
the centre of force. It now remains to determine those motions in eccen 
trical planes. For those authors who treat of the motion of heavy bodies 
used to consider the ascent and descent of such bodies, not only in a per 
pendicular direction, but at all degrees of obliquity upon any given planes ; 
and for the same reason we are to consider in this place the motions of 
bodies tending to centres by means of any forces whatsoever, when those 
bodies move in eccentrical planes. These planes are supposed to be 
perfectly smooth and polished, so as not to retard the motion of the bodies 
in the least. Moreover, in these demonstrations, instead of the planes upon 
which those bodies roll or slide, and which are therefore tangent planes to 
the bodies, I shall use planes parallel to them, in which the centres of the 
bodies move, and by that motion describe orbits. And by the same method 
I afterwards determine the motions of bodies performer 1 in curve superficies. 



SECTION X. 

Of the motion of bodies in given superficies, and of the reciprocal motion 
of fnnependulous bodies. 

PROPOSITION XL VI. PROBLEM XXXII. 

Any kind of centripetal force being supposed, and the centre of force , atfft 
any plane whatsoever in which the body revolves, being given, and tint 
quadratures of curvilinear figures being allowed; it is required to de 
termine the motion of a body going off from a given place., with a 
given velocity, in the direction of a given right line in, that plane. 




SEC. X.J OF NATURAL PHILOSOPHY- 183 

Let S be the centre of force, SC the 
least distance of that centre from the given 
plane, P a body issuing from the place P 
in the direction of the right line PZ, Q, 
the same body revolving in its trajectory, 
and PQ,R the trajectory itself which is 
required to be found, described in that 
given plane. Join CQ, Q.S, and if in Q,S 
we take SV proportional to the centripetal 
force with which the body is attracted to 
wards the centre S, and draw VT parallel 
to CQ, and meeting SC in T ; then will the force SV be resolved into 
two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ; of 
which ST aMracting the body in the direction of a line perpendicular to 
that plane, does not at all change its motion in that plane. But the action 
c f the other force TV, coinciding with the position of the plane itself, at 
tracts the body directly towards the given point C in that plane ; ad 
t icreftre causes the body to move in this plane in the same manner as if 
the force S F were taken away, and the body were to revolve in free space 
about the centre C by means of the force TV alone. But there being given 
the centripetal force TV with which the body Q, revolves in free space 
about the given centre C, there is given (by Prop. XLII) the trajectory 
PQ.R which the body describes ; the place Q,, in which the body will be 
found at any given time ; and, lastly, the velocity of the body in that place 
Q,. And so e contra. Q..E.I. 

PROPOSITION XLV1L THEOREM XV. 

Supposing the centripetal force to be proportional to t/ie distance of the 
body from the centre ; all bodies revolving in any planes whatsoever 
will describe ellipses, and complete their revolutions in equal times ; 
and those which move in right lines, running backwards and forwards 
alternately , will complete ttieir several periods of going and returning 
in the same times. 

For letting all things stand as in the foregoing Proposition, the force 
SV, with which the body Q, revolving in any plane PQ,R is attracted to 
wards the centre S, is as the distance SO. ; and therefore because SV and 
SQ,, TV and CQ, are proportional, the force TV with which the body is 
attracted towards the given point C in the plane of the orbit is as the dis 
tance CQ,. Therefore the forces with which bodies found in the plane 
PQ,R are attracted towaitis the point O, are in proportion to the distances 
equal to the forces with which the same bodies are attract-ed every way to 
wards the centre S ; and therefore the bodies will move in the same times, 
and in the same figures, in any plane PQR about the point C. n* they 



THE MATHEMATICAL PRINCIPLES [BOOK I. 

would do in free spaces about the centre S ; and therefore (by Cor. 2, Prop. 
X, ai d Cor. 2, Prop. XXXVIII.) they will in equal times either describe 
ellipses m that plane about the centre C, or move to and fro in right lines 
passing through the centre C in that plane; completing the same periods 
of time in all cases. Q.E.D. 

SCHOLIUM. 

The ascent and descent of bodies in curve superficies has a near relation 
to these motions we have been speaking of. Imagine curve lines to be de 
scribed on any plane, and to revolve about any given axes passing through 
the centre of force, and by that revolution to describe curve superficies ; and 
that the bodies move in such sort that their centres may be always found 
m those superficies. If those bodies reciprocate to and fro with an oblique 
ascent and descent, their motions will be performed in planes passing through 
tiie axis, and therefore in the curve lines, by whose revolution those curve 
superficies were generated. In those cases, therefore, it will be sufficient to 
consider thp motion in those curve lines. 

PROPOSITION XL VIII. THEOREM XVI. 

If a wheel stands npon the outside of a globe at right angles thereto, and 
revolving about its own axis goes forward in a great circle, the length 
of lite curvilinear path which any point, given in the perimeter of the 
wheel, hath described, since, the time that it touched the globe (which 
curvilinear path w~e may call the cycloid, or epicycloid), will be to double 
the versed sine of half the arc which since that time has touched the 
globe in passing over it, as the sn,m of the diameters of the globe and 
the wheel to the semi-diameter of the globe. 

PROPOSITION XLIX. THEOREM XVII. 

ff a wheel stand upon the inside of a concave globe at right angles there 
to, and revolving about its own axis go forward in one of the great 
circles of the globe, the length of the curvilinear path which any point, 
given in the perimeter of the wheel^ hath described since it toncJied the 
globe, imll be to the double of the versed sine of half the arc which in 
all that time has touched the globe in passing over it, as the difference 
of the diameters of the globe and the wheel to the semi-diameter of the 
globe. 

Let ABL be the globe. C its centre, BPV the wheel insisting thereon, 
E the centre of the wheel, B the point of contact, and P the given point 
in the perimeter of the wheel. Imagine this wheel to proceed in the great 
circle ABL from A through B towards L, and in its progress to revolve in 
such a manner that the arcs AB, PB may be always equal one to the other, 
:if;d the given point P in the peri meter of the wheel may describe in thf 



SEC. X.I 



OF NATURAL PHILOSOPHY. 

s 



185 



H 




mean time the curvilinear path AP. Let AP be the whole curvilinear 
path described since the wheel touched the globe in A, and the length cf 
this path AP will be to twice the versed sine of the arc |PB as 20 E to 
CB. For let the right line CE (produced if need be) meet the wheel in V, 
and join CP, BP, EP, VP ; produce CP, and let fall thereon the perpen 
dicular VF. Let PH, VH, meeting in H, touch the circle in P and V, 
and let PH cut YF in G, and to VP let fall the perpendiculars GI, HK. 
From the centre C with any interval let there be described the circle wow, 
cutting the right line CP in n t the perimeter of the wheel BP in o, and 
the curvilinear path AP in m ; and from the centre V with the interval 
Vo let there be described a circle cutting VP produced in q. 

Because the wheel in its progress always revolves about the point of con 
tact B. it is manifest that the right line BP is perpendicular to that curve line 
AP which the point P of the wheel describes, and therefore that the right 
line VP will touch this curve in the point P. Let the radius of the circle nmn 
be gradually increased or diminished so that at last it become equal to the 
distance CP ; and by reason of the similitude of the evanescent figure 
Pnn-mq, and the figure PFGVI, the ultimate ratio of the evanescent lined ae 
Pra, P//, Po, P<y, that is, the ratio of the momentary mutations of the curve 
AP, the right line CP, the circular arc BP, and the right line VP, will < 



iSS THE MATHEMATICAL PRINCIPLES [BOOK 1. 

the same as of the lines PV, PF, PG, PI, respectively. But since VF is 
perpendicular to OF, and VH to CV, and therefore the angles HVG, VCF 
equal: and the angle VHG (because the angles of the quadrilateral figure 
HVEP are right in V and P) is equal to the angle CEP, the triangles 
V HG, CEP will be similar ; and thence it will come to pass that as EP is 
to CE so is HG to HV or HP, and so KI to KP, and by composition or 
division as CB to CE so is PI to PK, and doubling the consequents asCB 
to 2CE so PI to PV, and so is Pq to Pm. Therefore the decrement of the 
line VP, that is, the increment of the line BY VP to the increment of the 
curve line AP is in a given ratio of CB to 2CE, and therefore (by Cor. 
Lena. IV) the lengths BY YP and AP, generated by those increments, arc 
in the same ratio. But if BY be radius, YP is the cosine of the angle BYP 
or -*BEP, and therefore BY YP is the versed sine of the same angle, and 
therefore in this wheel, whose radius is ^BV, BY YP will be double the 
versed sine of the arc ^BP. Therefore AP is to double the versed sine oi 
the arc ^BP as 2CE to CB. Q.E.D. 

The line AP in the former of these Propositions we shall name the cy 
cloid without the globe, the other in the latter Proposition the cycloid within 
the globe, for distinction sake. 

COR. 1. Hence if there be described the entire cycloid ASL, and the 
same be bisected in S, the len c th of the part PS will be to the length PV 
(which is the double of the sine of the angle YBP, when EB is radius) as 
2CE to CB, and therefore in a given ratio. 

COR. 2. And the length of the semi-perimeter of the cycloid AS will be 
equal to a right line which is to the dumeter of the wheel BY as 2CF 
toCB. 

PROPOSITION L. PROBLEM XXXIII. 

To cause a pendulous body to oscillate in a given cycloid. 

Let there be given within the globe QYS de- 
scribed with the centre C, the cycloid QRS, bi 
sected in R, and meeting the superficies of the 
globe with its extreme points Q and S on either 
hand. Let there be drawn CR birxcting the arc 
QS in O, and let it be produced to A in such 
sort that CA may be to CO as CO to CR. 
About the centre C, with the interval CA, let 
there be described an exterior globe DAF ; and 
within this globe, by a wheel whose diameter is 
AO, let there be described two semi-cycloids AQ,, 
AS, touching the interior globe in Q, and S, and meeting the exterior globe 
in A. From that point A, with a thread APT in length equal to the line 
AR, let the body T depend, and oscillate in such manner between the two 




SlCC. X.J OF NATURAL PHILOSOPHY. 187 

semi-cycloids AQ, AS, that, as often as the pendulum parts from the per 
pendicular AR, the upper part of the thread AP may be applied to that 
semi-cycloid APS towards which the motion tends, and fold itself round 
that curve line, as if it were some solid obstacle, the remaining part of the 
same thread PT which has not yet touched the semi-cycloid continuing 
straight. Then will the weight T oscillate in the given cycloid QRS. 

Q.E.F. 

For let the thread PT meet the cycloid QRS in T, and the circle QOS 
m V, and let 0V be drawn j and to the rectilinear part of the thread PT 
from the extreme points P and T let there be erected the perpendiculars 
BP, T W, meeting the right line CV in B and W. It is evident, from the 
construction and generation of the similar figures AS, SR, that those per 
pendiculars PB, TVV, cut off from CV the lengths VB, VVV equal the 
diameters of the wheels OA, OR. Therefore TP is to VP (which is dou 
ble the sine of the angle VBP when ^BV is radius) as BYV to BV, or AO 
-f-OR to AO, that is (since CA and CO, CO and CR ; and by division AO 
and OR are proportional), as CA + CO to CA, or, if BV be bisected in E, 
as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of the 
rectilinear part of the thread PT is always equal to the arc of the cycloid 
PS, and the whole thread APT is always equal to the half of the cycloid 
APS, that is (by Cor. 2, Prop. XLIX), to the length AR. And there 
fore contrariwise, if the string remain always equal to the length AR, the 
point T will always move in the given cycloid QRS. Q.E.D. 

COR. The string AR is equal to the semi-cycloid AS, and therefore has 
the same ratio to AC the semi-diameter of the exterior globe as the like 
semi-cycloid SR has to CO the semi-diameter of the interior globe. 

PROPOSITION LI. THEOREM XVIII. 

If a centripetal force tending on all sides to the centre C of a globe, be in 
all places as the distance of the place from the centre, and by this force 
alone acting upon it, the body T oscillate (in the manner above de 
scribed] in the perimeter of the cycloid QRS ; / say, that all the oscil 
lations, how unequal soever in tfiemselves, will be performed in equal 
times. 

For upon the tangent T W infinitely produced let fall the perpendicular 
CX, and join CT. Because the centripetal force with which the body T 
is impelled towards C is as the distance CT, let this (by Cor. 2, of the 
I ,aws) be resolved into the parts CX, TX, of which CX impelling the 
body directly from P stretches the thread PT, and by the resistance the 
rhread makes to it is totally employed, producing no other effect ; but the 
3ther part TX, impelling the body transversely or towards X, directly 
accelerates the motion in the cycloid. Then it is plain that the accelera 
tion of the body, proportional to this accelerating force, will bo every 



188 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1 




moment as the length TX, that is (because CV\ 
WV, and TX, TW proportional to them are given), 
as the length TW, that is (by Cor. 1, Prop. XLIX) 
as the length of the arc of the cycloid TR. If there 
fore two pendulums APT, Apt, be unequally drawn 
aside from the perpendicular AR, and let fall together, 
their accelerations will be always as the arcs to be de 
scribed TR, tR. But the parts described at the 
beginning of the motion are as the accelerations, thai 
is, as the wholes that are to be described at the be 
ginning, and therefore the parts which remain to be 
described, and the subsequent accelerations proportional to those parts, are 
also as the wholes, and so on. Therefore the accelerations, and consequently 
the velocities generated, and the parts described with those velocities, and 
the parts to be described, are always as the wholes ; and therefore the parts 
to be described preserving a given ratio to each other will vanish together, 
that is, the two bodies oscillating will arrive together at the perpendicular AR. 
And since on the other hand the ascent of the pendulums from the lowest place 
R through the same cycloidal arcs with a retrograde motion, is retarded in 
the several places they pass through by the same forces by which their de 
scent was accelerated : it is plain that the velocities of their ascent and de 
scent through the same arcs are equal, and consequently performed in equal 
times ; and, therefore, since the two parts of the cycloid RS and RQ lying 
on either side of the perpendicular are similar and equal, the two pendu 
lums will perform as well the wholes as the halves of their oscillations in 
the same times. Q.E.D. 

COR. The force with which the body T is accelerated or retarded in any 
place T of the cycloid, is to the whole weight of the same body in the 
highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR 

PROPOSITION LIL PROBLEM XXXIV. 

To define the velocities of the pendulums in the several places, and the 
times in which both the entire oscillations, and the several parts of them 
are performed. 

About any centre G, with the interval GH equal to 
the arc of the cycloid RS, describe a semi-circle HKM 
bisected by the semi-diameter GK. And if a centripe 
tal force proportional to the distance of the places from 
the centre tend to the centre G, and it be in the peri 
meter HIK equal to the centripetal force in the perime 
ter of the globe Q,OS tending towards its centre, and at 
the same time that the pendulum T is let fall from the 
highest place S, a body, as L, is let fall from H to G ; then because th< 





SEC. X.J OF NATURAL PHILOSOPHY. 189 

forces which act upon the bodies are equal at the be 
ginning, and always proportional to the spaces to be 
described TR, LG, and therefore if TR and LG are 
equal, are also equal in the places T and L, it is plain 
that those bodies describe at the beginning equal spaces M 
ST, HL, and therefore are still acted upon equally, and continue to describe 
equal spaces. Therefore by Prop. XXXVIII, the time in which the body 
describes the arc ST is to the time of one oscillation, as the arc HI the time 
in which the body H arrives at L, to the semi-periphery HKM, the time 
in which the body H will come to M. And the velocity of the pendulous 
body in the place T is to its velocity in the lowest place R, that is, the 
velocity of the body H in the place L to its velocity in the place G, or the 
momentary increment of the line HL to the momentary increment of the 
line HG (the arcs HI, HK increasing with an equable flux) as the ordinato 
LI to the radius GK. or as v/SR 2 Til 2 to SR. Hence, since in unequal 
oscillations there are described in equal time arcs proportional to the en 
tire arcs of the oscillations, there are obtained from the times given, both 
the velocities and the arcs described in all the oscillations universally. 
Which was first required. 

Let now any pendulous bodies oscillate in different cycloids described 
within different globes, whose absolute forces are also different ; and if the 
absolute force of any globe Q.OS be called V, the accelerative force with 
which the pendulum is acted on in the circumference of this globe, when it 
begins to move directly towards its centre, will be as the distance of the 
pendulous body from that centre and the absolute force of the globe con- 
junctly, that is, as CO X V. Therefore the lineola HY, which is as this 
accelerated force CO X V, will be described in a given time : and if there 
be erected the perpendicular YZ meeting the circumference in Z, the nascent 
arc HZ will denote that given time. But that nascent arc HZ is in the 
subduplicate ratio of the rectangle GHY, and therefore as v/GH X CO X V 
Whence the time of an entire oscillation in the cycloid Q,RS (it being as 
the semi-periphery HKM, w r hich denotes that entire oscillation, directly : 
and as the arc HZ which in like manner denotes a given time inversely) 
will be as GH directly and v/GH X CO X V inversely ; that is, because 

GH and SR are equal, as V nr , . or (by Cor. Prop. L,) as X/-TTVT- 

UU X V AO X V 

Therefore the oscillations in all globes and cycloids, performed with what 
absolute forces soever, are in a ratio compounded of the subduplicate ratio of 
the length of the string directly, and the subduplicate ratio of the distance 
between the point of suspension and the centre of the globe inversely, and 
the subduplicate ratio of the absolute force of the globe inversely also 
Q.E.I. 



t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1. 

COR. 1. Hence also the times of oscillating, falling, and revolving bodies 
may be compared among themselves. For if the diameter of the wheel 
with which the cycloid is described within the globe is supposed equal to 
the semi-diameter of the globe, the cycloid will become a right line passing 
through the centre of the globe, and the oscillation will be changed into a 
descent and subsequent ascent in that right line. Whence there is given 
both the time of the descent from any place to the centre, and the time equal 
to it in which the body revolving uniformly about the centre of the globe 
at any distance describes an arc of a quadrant For this time (by 
Case 2) is to the time of half the oscillation in any cycloid QJR.S as 1 to 

AR 
V AC 

COR. 2. Hence also follow what Sir Christopher Wren and M. Huygevs 
have discovered concerning the vulgar cycloid. For if the diameter of the 
globe be infinitely increased, its sphacrical superficies will be changed into a 
plane, and the centripetal force will act uniformly in the direction of lines 
perpendicular to that plane, and this cycloid of our s will become the same 
with the common cycloid. But in that case the length of the arc of the 
cycloid between that plane and the describing point will become equal to 
four times the versed sine of half the arc of the wheel between the same 
plane and the describing point, as was discovered by Sir Christopher Wren. 
And a pendulum between two such cycloids will oscillate in a similar and 
equal cycloid in equal times, as M. Huygens demonstrated. The descent 
of heavy bodies also in the time of one oscillation will be the same as M. 
Huygens exhibited. 

The propositions here demonstrated are adapted to the true constitution 
of the Earth, in so far as wheels moving in any of its great circles will de 
scribe, by the motions of nails fixed in their perimeters, cycloids without the 
globe ; and pendulums, in mines and deep caverns of the Earth, must oscil 
late in cycloids within the globe, that those oscillations may be performed 
in equal times. For gravity (as will be shewn in the third book) decreases 
in its progress from the superficies of the Earth ; upwards in a duplicate 
ratio of the distances from the centre of the Earth ; downwards in a sim 
ple ratio of the same. 

PROPOSITION LIII. PROBLEM XXXV. 

Granting the quadratures of curvilinear figures, it is required to find 
the forces with which bodies moving in given curve lines may always 
perform their oscillations in equal times. 

Let the body T oscillate in any curve line STRQ,, whose axis is AR 
passing through the centre of force C. Draw TX touching that curve in 
any place of the body T, and in that tangent TX take TY equal to the 
arc TR. The length of that arc is known from the common methods used 



SEC. X. 



OF NATURAL PHILOSOPHY. 



191 




for the quadratures of figures. From the point Y 
draw the right line YZ perpendicular to the tangent. 
Draw CT meeting that perpendicular in Z, and the 
centripetal force will be proportional to the right line 
TZ. Q.E.I. 

For if the force with which the body is attracted 
from T towards C be expressed by the right line TZ 
taken proportional to it, that force will be resolved 
into two forces TY, YZ, of which YZ drawing the 
body in the direction of the length of the thread PT, 
docs not at all change its motion ; whereas the other 
force TY directly accelerates or retards its mction in the curve STRQ. 
Wherefore since that force is as the space to be described TR, the acceler 
ations or retardations of the body in describing two proportional parts (u 
greater arid a less) of two oscillations, will be always as those parts, and 
therefore will cause those parts to be described together. But bodies which 
continually describe together parts proportional to the wholes, will describe 
the wholes together also. Q,.E.l). 

COR. 1. Hence if the body T, hanging by a rectilinear thread 
AT from the centre A, describe the circular arc STRQ,, 
and in the mean time be acted on by any force tending 
downwards with parallel directions, which is to the uni 
form force of gravity as the arc TR to its sine TN, the 
times of the several oscillations will be equal. For because 
TZ, AR are parallel, the triangles ATN, ZTY are similar ; and there 
fore TZ will be to AT as TY to TN ; that is, if the uniform force of 
gravity be expressed by the given length AT, the force TZ. by which the 
oscillations become isochronous, will be to the force of gravity AT, as the 
arc TR equal to TY is to TN the sine of that arc. 

COR. 2. And therefore in clocks, if forces were impressed by some ma 
chine upon the pendulum which preserves the motion, and so compounded 
with the force of gravity that the whole force tending downwards should 
be always as a line produced by applying the rectangle under the arc TR 
and the radius AR to the sine TN, all the oscillations will become 
isochronous. 

PROPOSITION LIV. PROBLEM XXX YI. 

Granting the quadratures of curvilinear figures, it is required to find 
the times in which bodies by means of any centripetal force will descend 
or ascend in any curve lines described in, a plane passing through the 
centre of force. 
Let the body descend from any place S, and move in any curve ST/R 

given in a plane passing through the centre of force C. Join CS, and lei 




192 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1 





Q it be divided into innumerable equal parts, and let 
Dd be one of those parts. From the centre C, with 
the intervals CD, Cd, let the circles DT, dt be de 
scribed, meeting the curve line ST*R in T and t. 
And because the law of centripetal force is given. 
and also the altitude CS from which the body at 
first fell, there will be given the velocity of the body 
in any other altitude CT (by Prop. XXXIX). But 
the time in which the body describes the lineola Tt 
is as the length of that lineola, that is, as the secant 
of the angle /TC directly, and the velocity inversely. 
Lei, the ordinate DN, proportional to this time, be made perpendicular to 
the right line CS at the point D, and because Dd is given, the rectangle 
Dd X DN, that is, the area DNwc?, will be proportional to the same time. 
Therefore if PN/?, be a curve line in which the point N is perpetually found, 
and its asymptote be the right line SQ, standing upon the line CS at right 
angles, the area SQPJN D will be proportional to the time in which the body 
in its descent hath described the line ST ; and therefore that area being 
found, the time is also given. Q.E.I. 



PROPOSITION LV. THEOREM XIX. 

If a body move in any curve superficies, whose axis passes through the 
centre of force, and from the body a perpendicular be let fall iipon the 
axis \ and a line parallel and equal thereto be drawn from any given 
point of the axis ; I say, that this parallel line will describe an area 
proportional to the time. 

Let BKL be a curve superficies, T a body 
revolving in it, STR a trajectory which the 
body describes in the same, S the beginning 
of the trajectory, OMK the axis of the curve 
superficies, TN a right line let fall perpendic 
ularly from the body to the axis ; OP a line 
parallel and equal thereto drawn from the 
given point O in the axis ; AP the orthogra 
phic projection of the trajectory described by 
the point P in the plane AOP in which the 
revolving line OP is found : A the beginning 
of that projection, answering to the point S ; 
TO a right line drawn from the body to the centre ; TG a part thereof 
proportional to the centripetal force with which the body tends towards the 
centre C ; TM a right line perpendicular to the curve superficies ; TI a 
part thereof proportional to the force of pressure with which the body urges 




SEC. X.] 



OF NATURAL PHILOSOPHY. 



193 



the superficies, and therefore with which it is again repelled by the super 
ficies towards M ; PTF a right line parallel to the axis and passing through 
the body, and OF, IH right lines let fall perpendicularly from the points 
G and I upon that parallel PHTF. I say, now. that the area AGP, de 
scribed by the radius OP from the beginning of the motion, is proportional 
to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re 
solved into the forces TF, FG ; and the force TI into the forces TH, HI ; 
but the forces TF, TH, acting in the direction of the line PF perpendicular 
to the plane AOP, introduce no change in the motion of the body but in a di 
rection perpendicular to that plane. Therefore its motion, so far as it has 
the same direction with the position of the plane, that is, the motion of the 
point P, by which the projection AP of the trajectory is described in that 
plane, is the same as if the forces TF, TH were taken away, and the body 
wei e acted on by the forces FG, HI alone ; that is, the same as ,f the body 
were to describe in the plane AOP the curve AP by means of a centripetal 
force tending to the centre O, and equal to the sum of the forces FG and 
HI. But with such a force as that (by Prop. 1) the area AOP will be de 
scribed proportional to the time. Q.E.D. 

COR. By the same reasoning, if a body, acted on by forces tending to 
two or more centres in any the same right line CO, should describe in a 
free space any curve line ST, the area AOP would be always proportional 
to the time. 

PROPOSITION LVI. PROBLEM XXXVII. 

Granting the quadratures of curvilinear figures, and supposing that 
there are given both the law of centripetal force tending to a given cen 
tre, and the curve superficies whose axis passes through that centre ; 
it is required to find the trajectory which a body will describe in that 
superficies, when going off from a given place with a given velocity, 
and in a given direction in that superficies. 
The last construction remaining, let the 
body T go from the given place S, in the di 
rection of a line given by position, and turn 
into the trajectory sought STR, whose ortho 
graphic projection in the plane BDO is AP. 
And from the given velocity of the body in 
the altitude SC, its velocity in any other al 
titude TC will be also given. With that 
velocity, in a given moment of time, let the 
body describe the particle Tt of its trajectory, 
and let P/? be the projection of that particle 
described in the plane AOP. Join Op, and 
a little circle being described upon the curve superficies about the centre T 

13 




194 THE MATHEMATICAL PRINCIPLES [BOOK I 

with the interval TV let the projection of that little circle in the plane AOP 
be the ellipsis pQ. And because the magnitude of that little circle T/, and 
TN or PO its distance from the axis CO is also given, the ellipsis pQ, will 
be given both in kind and magnitude, as also its position to the right line 
PO. And since the area PO/? is proportional to the time, and therefore 
given because the time is given, the angle POp will be given. And thence 
will be given jo the common intersection of the ellipsis and. the right line 
Op, together with the angle OPp, in which the projection APp of the tra 
jectory cuts the line OP. But from thence (by conferring Prop. XLI, with 
Us 2d Cor.) the mariner of determining the curve APp easily appears. 
Then from the several points P of that projection erecting to the plane 
AOP, the perpendiculars PT meeting the curve superficies in T, there will 
be iven the several points T of the trajectory. Q.E.I. 



SECTION XL 

f f the motions of bodies tending to each other with centripetal forces. 
I have hitherto been treating of the attractions of bodies towards an im 
movable centre; though very probably there is no such thing existent in 
nature. For attractions are made towards bodies, and the actions of the 
bodies attracted and attracting are always reciprocal and equal, by Law III ; 
BO that if there are two bodies, neither the attracted nor the attracting body 
is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it 
were mutually attracted, revolve about a common centre of gravity. And 
if there be more bodies, which are either attracted by one single one which 
is attracted by them again, or which all of them, attract each other mutu 
ally , these bodies will be so moved among themselves, as that their common 
centre of gravity will either be at rest, or move uniformly forward in a 
right line. I shall therefore at present go on to treat of the motion of 
bodies mutually attracting each other ; considering the centripetal forces 
as attractions ; though perhaps in a physical strictness they may more truly 
be called impulses. But these propositions are to be considered as purely 
mathematical; and therefore, laying aside all physical considerations, I 
make use of a familiar way of speaking, to make myself the more easily 
understood by a mathematical reader. 

PROPOSITION LVII. THEOREM XX. 

Two bodies attracting each other mutually describe similar figures about 

their common centre of gravity, and about each other mutually. 

For the distances of the bodies from their common centre of gravity are 

leciprocally as the bodies; and therefore in a given ratio to each other: 

*nd thence, bv composition of ratios, in a given ratio to the whole distance 



SEC. XL] OF NATURAL PHILOSOPHY. 195 

between the bodies. Now these distances revolve about their common term 
with an equable angular motion, because lying in the same right line they 
never change their inclination to each other mutually But right lines 
that are in a given ratio to each other, and revolve about their terms with 
an equal angular motion, describe upon planes, which either rest with 
those terms, or move with any motion not angular, figures entirely similar 
round those terms. Therefore the figures described by the revolution ot 
these distances are similar. Q.E.D. 



PROPOSITION LVIll. THEOREM XXI. 

If two bodies attract each other mutually with forces of any kind, and 
in the mean time revolve about the common centre of gravity ; I say, 
that, by the same forces, there may be described round either body un 
moved ajigure similar and equal to the figures ivhich the bodies so 
moving describe round each other mutually. 
Let the bodies S and P revolve about their common centre of gravity 

C, proceeding from S to T, and from P to Q,. From the given point s lot 





there be continually drawn sp, sq, equal and parallel to SP, TQ, ; and the 
;urve pqv, which the point p describes in its revolution round the immovable 
point s, will be similar and equal to the curves which the bodies S and P 
describe about each other mutually ; and therefore, by Theor. XX, similar 
to the curves ST and PQ,V which the same bodies describe about their 
common centre of gravity C and that because the proportions of the lines 
SC. CP, and SP or sp, to each other, are given. 

CASE 1. The common centre of gravity C (by Cor. 4, of the Laws of Mo 
tion) is either at rest, or moves uniformly in a right line. Let us first 
suppose it at rest, and in s and p let there be placed two bodies, one im 
movable in s, the other movable in p, similar and equal to the bodies S arid 
P. Then let the right lines PR and pr touch the curves PQ, and pq ki P 
and p, and produce CQ, and sq to R and r. And because the figures 
CPRQ, sprq are similar, RQ, will be to rq as CP to sp, and therefore in a 
given ratio. Hence if the force with which the body P is attracted to 
wards the body S, and by consequence towards the intermediate point the 
centre C, were to the force with which the body p is attracted towards the 
Centre 5. in the same given ratio, these forces would in equal times attract 



196 THE MATHEMATICAL PRINCIPLES |BoOK 1 

the bodies from the tangents PR, pr to the arcs PQ, pq, through the in 
tervals proportional to them RQ,, rq ; and therefore this last force (tending 
to s) would make the body p revolve in the curve pqv, which would becomr 
similar to the curve PQV, in which the first force obliges the body P i( 
revolve ; and their revolutions would be completed in the same timeg 
But because those forces are not to each other in the ratio of CP to sp, bu; 
(by reason of the similarity and equality of the bodies S and s, P and / 
and the equality of the distances SP, sp) mutually equal, the bodies ii 
equal times will be equally drawn from the tangents; and therefore tLV 
the body p may be attracted through the greater interval rq, there is re 
quired a greater time, which will be in the subduplicate ratio of the inter 
vals ; because, by Lemma X, the spaces described at the very beginning ol 
the motion are in a duplicate ratio of the times. Suppose, then the velocity 
of the body p to be to the velocity of the body P in a subduplicate ratio of 
the distance sp to the distance CP, so that the arcs pq, PQ, which are in a 
simple proportion to each other, may be described in times that are in n 
subduplicate ratio of the distances ; and the bodies P, p, always attracted 
by equal forces, will describe round the quiescent centres C and s similar 
figures PQV, pqv, the latter of which pqv is similar and equal to the figure 
ivhich the body P describes round the movable body S. Q.E.I) 

CASE 2. Suppose now that the common centre of gravity, together with 
the space in which the bodies are moved among themselves, proceeds uni 
formly in a right line ; and (by Cor. 6, of the Laws of Motion) all the mo 
tions in this space will be performed in the same manner as before ; and 
therefore the bodies will describe mutually about each other the same fig 
ures as before, which will be therefore similar and equal to the figure pqv. 
Q.E.D. 

COR. 1. Hence two bodies attracting each other with forces proportional 
to their distance, describe (by Prop. X) both round their common centre ol 
gravity, and round each other mutually concentrical ellipses ; and, vice 
versa, if such figures are described, the forces are proportional to the dis 
tances. 

COR. 2. And two bodies, whose forces are reciprocally proportional to 
the square of their distance, describe (by Prop. XI, XII, XIII), both round 
their common centre of gravity, and round each other mutually, conic sec 
tions having their focus in the centre about which the figures are described. 
And, vice versa, if such figures are described, the centripetal forces are re 
ciprocally proportional to the squares of the distance. 

COR. 3. Any two bodies revolving round their common centre of gravity 
describe areas proportional to the times, by radii drawn both to that centre 
and to each other mutually- 



>EC. XL] OF NATURAL PHILOSOPHY. 197 

PROPOSITION LIX. THEOREM XXII. 

The periodic time of two bodies S and P revolving round their common 
centre of gravity C,is to the periodic time of one of the bwlies 1? re 
volving round the other S remaining unmoved, and describing a fig 
ure similar and equal to those which the bodies describe about each 
other mutually r , in a subduplicate ratio of the other body S to the sii/rn 
of the bodies S -f P. 

For, by the demonstration of the last Proposition, the times in which 
any similar arcs PQ and pq are described are in a subduplicate ratio of the 
distances CP and SP, or sp, that is, in a subduplicate ratio of the ody S 
to the sum of the bodies S + P. And by composition of ratios, the sums 
of the times in which all the similar arcs PQ and pq are described, that is, 
the whole times in which the whole similar figures are described are in the 
same subduplicate ratio. Q.E.D. 

PROPOSITION LX. THEOREM XXIII. 

If two bodies S and P, attracting each other with forces reciprocally pro 
portional to the squares of their distance, revolve about their common 
centre of gravity ; I say, that the principal axis of the ellipsis which 
either of the bodies, as P, describes by this motion about the other S, 
will be to the principal axis of the ellipsis, which the same body P may 
describe in the same periodical time about the other body S quiescent, 
as the sum of the two bodies S + P to the first of two mean, propor 
tionals between that sum and the other body S. 

For if the ellipses described were equal to each other, their periodic times 
by the last Theorem would be in a subduplicate ratio of the body S to the 
sum of the bodies S 4- P. Let the periodic time in the latter ellipsis be 
diminished in that ratio, and the periodic times will become equal ; but, 
by Prop. XV, the principal axis of the ellipsis will be diminished in a ratio 
sesquiplicate to the former ratio ; that is, in a ratio to which the ratio of 
S to S 4- P is triplicate ; and therefore that axis will be to the principal 
axis of the other ellipsis as the first of two mean proportionals between S 
-f- P and S to S 4- P. And inversely the principal axis of the ellipsis de 
scribed about the movable body will be to the principal axis of that described 
round the immovable as S + P to the first of two mean proportionals be 
tween S 4- P and S. Q.E.D. 

PROPOSITION LXI. THEOREM XXIV. 

If two bodies attracting each other with any kind of forces, and not 
otherwise agitated or obstructed, are moved in any manner whatsoever, 
those motions will be the same as if they did not at all attract each 
other mutually, but were both attracted with the same forces by a third 
body placed in their common centre of gravity ; and the law of the 



198 THE MATHEMATICAL PRINCIPLES [BOOK I 

attracting Jones will be the sam# in respect of the distance of the. 

bodies from, the common centre, as in respect of the distance between 

the two bodies. 

For those forces with which the bodies attract each other mutually, by 
tending to the bodies, tend also to the common centre of gravity lying di 
rectly between them ; and therefore are the same as if they proceeded from 
an intermediate body. QJG.D. 

And because there is given the ratio of the distance of either body from 
that common centre to the distance between the two bodies, there is given, 
of course, the ratio of any power of one distance to the same power of the 
. ther distance ; and also the ratio of any quantity derived in any manner 
from one of the distances compounded any how with given quantities, to 
another quantity derived in like manner from the other distance, and as 
many given quantities having that given ratio of the distances to the first 
Therefore if the force with which one body is attracted by another be di 
rectly or inversely as the distance of the bodies from each other, or as any 
power of that distance ; or, lastly, as any quantity derived after any man 
ner from that distance compounded with given q-uantities ; then will the 
same force with which the same body is attracted to the common centre of 
gravity be in like manner directly or inversely as the distance of the at 
tracted body from the common centre, or as any power of that distance ; 
or, lastly, as a quantity derived in like sort from that distance compounded 
with analogous given quantities. That is, the law of attracting force will 
be the same with respect to both distances. Q,.E.D. 

PROPOSITION LXII. PROBLEM XXXVIII. 

To determine the motions of two bodies which attract each other with 
forces reciprocally proportional to the squares of the distance between 
them, aflid are, let fall from given places. 

The bodies, by the last Theorem, will be moved in the same manner as 
if they were attracted by a third placed in the common centre of their 
gravity ; and by the hypothesis that centre will be quiescent at the begin 
ning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will 
be always quiescent. The motions of the bodies are therefore to be deter 
mined (by Prob. XXV) in the same manner as if they were impelled by 
forces tending to that centre: and then we shall have the motions of the 
bodies attracting each other mutually. Q.E.I. 

PROPOSITION LXIII. PROBLEM XXXIX. 

To determine the motions of two bodies attracting each other with forces 
reciprocally proportional to the squares of their distance, and going 
off from given places in, given directions with given velocities. 
The motions of the bodies at the beginning being given, there is given 



SEC. XL] OF NATURAL PHILOSOPHY. 1 % 

also the uniform motion of the common centre of gravity, and the motion 
of the space which moves along with this centre uniformly in a right line, 
and also the very first, or beginning motions of the bodies in respect of this 
space. Then (by Cor. 5, of the Laws, and the last Theorem) the subse 
quent motions will be performed in the same manner in that space, as if 
that space together with the common centre of gravity were at rest, and as 
if the bodies did not attract each other, but were attracted by a third body 
placed in that centre. The motion therefore in this movable space of each 
body going off from a given place, in a given direction, with a given velo 
city, and acted upon by a centripetal force tending to that centre, is to be 
determined by Prob. IX and XXVI, and at the same time will be obtained 
the motion of the other round the same centre. With this motion com 
pound the uniform progressive motion of the entire system of the space and 
the bodies revolving in it, and there will be obtained the absolute motion 
of the bodies in immovable space. Q..E.I. 

PROPOSITION LXIV. PROBLEM XL. 

Supposing forces with which bodies mutually attract each other to in 
crease in a simple ratio of their distances from the centres ; it is ro- 
quired to find the motions of several bodies among themselves. 
Suppose the first two bodies T and L 
to have their common centre of gravity in 
L). These, by Cor. 1, Theor. XXI, will S 
describe ellipses having their centres in D, 
the magnitudes of which ellipses are 

known by Prob. V. J- -- \- ? L 

Let now a third body S attract the two 
former T and L with the accelerative forces ST, SL, and let it be attract 
ed again by them. The force ST (by Cor. 2, of the Laws of Motion) is 
resolved into the forces SD, DT ; and the force SL into the forces SD and 
DL. Now the forces DT, DL. which are as their sum TL, and therefore 
as the accelerative forces with which the bodies T and L attract each other 
mutually, added to the forces of the bodies T and L, the first to the first, 
and the last to the last, compose forces proportional to the distances DT 
and DL as before, but only greater than those former forces : and there 
fore (by Cor. 1, Prop. X, and Cor. l,and 8, Prop. IV) they will cause those 
bodies to describe ellipses as before, but with a swifter motion. The re 
maining accelerative forces SD and DL, by the motive forces SD X Tand 
SD X L, which are as the bodies attracting those bodies equally and in the 
direction of the lines TI, LK parallel to DS, do not at all change their situ 
ations with respect to one another, but cause them equally to approach to 
the line IK ; which must be imagined drawn through the middle of the 
body S, and perpendicular to the line DS. But that approach to the line 




200 THE MATHEMATICAL PRINCIPLES [BoOK I. 

IK will be hindered by causing the system of the bodies T and L on one 
side, and the body S on the other, with proper velocities, to revolve round 
the common centre of gravity C. With such a motion the body S, because 
the sum of the motive forces SD X T and SD X L is proportional to the 
distance OS, tends to the centre C, will describe an ellipsis round the same 
centre C; and the point D, because the lines CS and CD are proportional, 
will describe a like ellipsis over against it. But the bodies T and L, at 
tracted by the motive forces SD X T and SD X L, the first by the first, 
and the last by the last, equally and in the direction of the parallel lines TI 
and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion) 
continue to describe their ellipses round the movable centre D, as before. 
Q.E.I. 

Let there be added a fourth body V, and, by the like reasoning, it will 
be demonstrated that this body and the point C will describe ellipses about 
the common centre of gravity B ; the motions of the bodies T, L, and S 
round the centres D and C remaining the same as before ; but accelerated. 
Arid by the same method one may add yet more bodies at pleasure. Q..E.I. 
^This would be the case, though the bodies T and L attract each other 
mutually with accelerative forces either greater or less than those with 
which they attract the other bodies in proportion to their distance. Let 
all the mutual accelerative attractions be to each other as the distances 
multiplied into the attracting bodies ; and from what has gone before it 
will easily be concluded that all the bodies will describe different ellipses 
with equal periodical times about their common centre of gravity B, in an 
immovable plane. Q.E.I. 

PROPOSITION LXV. THEOREM XXV. 

Bodies, whose forces decrease in a duplicate ratio of their distances from 
their centres, may move among" themselves in ellipses ; and by radii 
drawn to the foci may describe areas proportional to the times very 
nearly. 

In the last Proposition we demonstrated that case in which the motions 
will be performed exactly in ellipses. The more distant the law of the 
forces is from the law in that case, the more will the bodies disturb each 
other s motions ; neither is it possible that bodies attracting each other 
mutually according to the law supposed in this Proposition should move 
exactly in ellipses, unless by keepirg a certain proportion of distances from 
each other. However, in the following crises the orbits will not much dif 
fer from ellipses. 

CASE I. Imagine several lesser bodies to revolve about some very great 
one at different distances from it, and suppose absolute forces tending to 
rvery one of the bodies proportional to each. And because (by Cor. 4, ol 
the I aws) the common centre of gravity of them all is either at rest, 01 



iSEC. XL] OF NATURAL PHILOSOPHY. 20 i 

moves uniformly forward in a right line, suppose the lesser bodies so small 
that the groat body may be never at a sensible distance from that centre ; 
and then the great body will, without any sensible error, be either at rest, 
or move uniformly forward in a right line; and the lesser will revolve 
about that great one in ellipses, and by radii drawn thereto will describe 
areas proportional to the times ; if we except the errors that may be intro 
duced by the receding of the great body from the common centre of gravity, 
or by the mutual actions of the lesser bodies upon each other. But the 
lesser bodies may be so far diminished, as that this recess and the mutual 
actions of the bodies on each other may become less than any assignable; 
and therefore so as that the orbits may become ellipses, and the areas an 
swer to the times, without any error that is not less than any assignable. 
Q.E.O. 

CASE 2. Let us imagine a system of lesser bodies revolving about a very 
great one in the manner just described, or any other system of two bodies 
revolving about each other to be moving uniformly forward in a right line, and 
in the mean time to be impelled sideways by the force of another vastly greater 
body situate at a great distance. And because the equal accelerative forces 
with which the bodies are impelled in parallel directions do not change the 
situation of the bodies with respect to each other, but only oblige the whole 
system to change its place while the parts still retain their motions among 
themselves, it is manifest that no change in those motions of the attracted 
bodies can arise from their attractions towards the greater, unless by the 
inequality of the accelerative attractions, or by the inclinations of the lines 
towards each other, in whose directions the attractions are made. Suppose, 
therefore, all the accelerative attractions made towards the great body 
to be among themselves as the squares of the distances reciprocally ; and 
then, by increasing the distance of the great body till the differences of fhe 
right lines drawn from that to the others in respect of their length, and the 
inclinations of those lines to each other, be less than any given, the mo 
tions of the parts of the system will continue without errors that are not 
less than any given. And because, by the small distance of those parts from 
each other, the whole system is attracted as if it were but one body, it will 
therefore be moved by this attraction as if it were one body ; that is, its 
centre of gravity will describe about the great bod/ one of the conic sec 
tions (that is, a parabola or hyperbola when the attraction is but languid 
and an ellipsis when it is more vigorous) ; and by radii drawn thereto, it 
will describe areas proportional to the times, without any errors but thos 
which arise from the distances of the parts, which are by the supposition 
exceedingly small, and may be diminished at pleasure. Q,.E.O. 

By a like reasoning one may proceed to more compounded cases in in- 
finitum. 

COR 1. In the second Case, the nearer the very great body approaches to 



^0 ^ THE MATHEMATICAL PRINCIPLES [CoOK I 

the system of two or more revolving bodies, the greater will the pertur 
bation be of the motions of the parts of the system among themselves; be 
cause the inclinations of the lines drawn from that great body to those 
parts become greater ; and the inequality of the proportion is also greater. 

COR. 2. But the perturbation will be greatest of all, if we suppose the 
uccelerative attractions of the parts of the system towards the greatest body 
of all are not to each other reciprocally as the squares of the distances 
from that great body ; especially if the inequality of this proportion be 
greater than the inequality of the proportion of the distances from the 
great body. For if the accelerative force, acting in parallel directions 
and equally, causes no perturbation in the motions of the parts of the 
system, it must of course, when it acts unequally, cause a perturbation some 
where, which will be greater or less as the inequality is greater or less. 
The excess of the greater impulses acting upon some bodies, and not acting 
upon others, must necessarily change their situation among themselves. And 
this perturbation, added to the perturbation arising from the inequality 
and inclination of the lines, makes the whole perturbation greater. 

COR. *. Hence if the parts of this system move in ellipses or circles 
without any remarkable perturbation, it is manifest that, if they are at all 
impelled by accelerative forces tending to any other bodies, the impulse is 
very weak, or else is impressed very near equally and in parallel directions 
upon all of them. 

PROPOSITION LXVL THEOREM XXVI. 

Tf three bodies whose forces decrease in a duplicate ratio of the distances 
attract each other mutually ; and the accelerative attractions of any 
two towards the third be between themselves reciprocally as the squares, 
of the distances ; and the two least revolve about the greatest ; I say, 
that the interior of the tivo revolving bodies will, by radii drawn to the 
innermost and greatest, describe round thai body areas more propor 
tional to the times, and a figure more approaching to that of an ellip 
sis having its focus in the point of concourse of the radii, if that great 
body be agitated by those attractions, than it would do if lhat great 
body were not attracted at all by the lesser, but remained at rest ; or 
than it would if that great body were very much more or very much 
less attracted, <>r very much more or very much less agitated, by the 
attractions. 
This appears plainly enough from the demonstration of the second 

Corollary of tl.e foregoing Proposition; but it may be made out after 

this manner by a way of reasoning more distinct and more universally 

convincing. 

CASE 1. Let the lesser bodies P and S revolve in the same plane about 

the greatest body T, the body P describing the interior orbit PAB, and S 



SEC. XI.J OF NATURAL PHILOSOPHY. 203 

the exterior orbit ESE. Let SK be the mean distance of the bodies P and 

S ; and let the accelerative attraction of the body P towards S, at that 

mean distance, be expressed by that line SK. Make SL to SK as the 

E C 




square of SK to the square of SP, and SL will be the accelerative attrac 
tion of the body P towards S at any distance SP. Join PT, and draw 
LM parallel to it meeting ST in M; and the attraction SL will be resolv 
ed (by Cor. 2. of the Laws of Motion) into the attractions SM, LM. And 
so the body P will be urged with a threefold accelerative force. One of 
these forces tends towards T, and arises from the mutual attraction of the 
bodies T and P. By this force alone the body P would describe round the 
body T, by the radius PT, areas proportional to the times, and an 
ellipsis whose focus is in the centre of the body T ; and this it would do 
whether the body T remained unmoved, or whether it were agitated by that 
attraction. This appears from Prop. XI, and Cor. 2 and 3 of Theor. 
XXI. The other force is that of the attraction LM, which, because it 
tends from P to T, will be superadded to and coincide with the former 
force ; and cause the areas to be still proportional to the times, by Cor. 3, 
Theor. XXI. But because it is not reciprocally proportional to the square 
of the distance PT, it will compose, when added to the former, a force 
varying from that proportion : which variation will be the greater by how 
much the proportion of this force to the former is greater, cceteris paribus. 
Therefore, since by Prop. XI, and by Cor. 2, Theor. XXI, the force with 
which the ellipsis is described about the focus T ought to be directed to 
that focus, and to be reciprocally proportional to the square of the distance 
PT, that compounded force varying from that proportion will make the 
orbit PAB vary from the figure of an ellipsis that has its focus in the point 
I 1 ; and so much the more by how much the variation from that proportion 
is greater ; and by consequence by how much the proportion of the second 
force LM to the first force is greater, cceteris paribus. But now the third 
force SM, attracting the body P in a direction parallel to ST, composes with 
the other forces a new force which is no longer directed from P to T : and which 
varies so much more from this direction by how much the proportion of this 
third force to the other forces is greater, cceteris paribus ; arid therefore causes 
the body P to describe, by the radius TP, areas no longer proportional to the 
times : and therefore makes the variation from that proportionality so much 
greater by how much the proportion of this force to the others is greater. 
But this third force will increase the variation of the orbit PAB from th* 



THE MATHEMATICAL PRINCIPLES [BOOK 1 

elliptical figure before-mentioned upon two accounts ; first because that 
force is not directed from P to T ; and, secondly, because it is not recipro 
cally proportional to the square of the distance PT. These things being 
premised, it is manifest that the areas are then most nearly proportional to 
the times, when that third force is the least possible, the rest preserving 
their former quantity ; and that the orbit PAB does then approach nearest 
to the elliptical figure above-mentioned, when both the second and third, 
but especially the third force, is the least possible; the first force remain 
ing in its former quantity. 

Let the accelerative attraction of the body T towards S be expressed by 
the line SN ; then if the accelerative attractions SM and SN were equal, 
these, attracting the bodies T and P equally and in parallel directions 
would not at all change their situation with respect to each other. The mo 
tions of the bodies between themselves would be the same in that case as if 
those attractions did not act at all, by Cor. 6, of the Laws of Motion. And, 
by a like reasoning, if the attraction SN is less than the attraction SM, it 
will take away out of the attraction SM the part SN, so that there will re 
main only the part (of the attraction) MN to disturb the proportionality of 
the areas and times, and the elliptical figure of the orbit. And in like 
manner if the attraction SN be greater than the attraction SM, the pertur 
bation of the orbit and proportion will be produced by the difference MN 
alone. After this manner the attraction SN reduces always the attraction 
SM to the attraction MN, the first and second attractions rema ning per 
fectly unchanged ; and therefore the areas and times come then nearest to 
proportionality, and the orbit PAB to the above-mentioned elliptical figure, 
when the attraction MN is either none, or the least that is possible; that 
is, when the accelerative attractions of the bodies P and T approach as near 
as possible to equality ; that is, when the attraction SN is neither none at 
all, nor less than the least of all the attractions SM, but is, as it were, a 
mean between the greatest and least of all those attractions SM, that is, 
not much greater nor much less than the attraction SK. Q.E.D. 

CASE 2. Let now the lesser bodies P. S, revolve about a greater T in dif 
ferent planes ; and the force LM, acting in the direction of the line PT 
situate in the plane of the orbit PAB, will have the same effect as before ; 
neither will it draw the body P from the plane of its orbit. But the other 
force NM acting in the direction of a line parallel to ST (and which, there 
fore, when the body S is without the line of the nodes is inclined to the 
plane of the orbit PAB), besides the perturbation of the motion just now 
spoken of as to longitude, introduces another perturbation also as to latitude, 
attracting the body P out of the plane of its orbit. And this perturbation, 
in any given situation of the bodies P and T to each other, will be as the 
generating force MN ; and therefore becomes least when the force MN is 
least, that is (as was just now shewn), where the attraction SN is not nrirb 
greater nor much less than the attraction SK. Q.E.D. 



SK-C. XL] OF NATURAL PHILOSOPHY. 205 

COR. 1. Hence it may be easily collected, that if several less bodies P 
8, R, &c. ; revolve about a very great body T, the motion of the innermost 
revolving body P will be least disturbed by the attractions of the others. 
when the great body is as well attracted and agitated by the rest (accord 
ing to the ratio of the accelerative forces) as the rest are by each other 
mutually. 

COR. 2. In a system of three bodies, T, P, S, if the accelerative attrac 
tions of any two of them towards a third be to each other reciprocally as the 
squares of the distances, the body P, by the radius PT, will describe its area 
about the body T swifter near the conjunction A and the opposition B than it 
will near the quadratures C arid D. For every force with which the body P 
is acted on and the body T is not, and which does not act in the direction of 
the line PT, does either accelerate or retard the description of the area, 
according as it is directed, whether in consequentia or in cwtecedentia. 
Such is the force NM. This force in the passage of the body P frcm C 
to A is directed in consequentia to its motion, and therefore accelerates 
it; then as far as D in atttecedentia, and retards the motion; then in, con 
sequentia as far as B ; and lastly in antecedentia as it moves from B to C. 

COR. 3. And from the same reasoning it appears that the body P ccBteris 
paribuSj moves more swiftly in the conjunction and opposition than in the 
quadratures. 

COR. 4. The orbit of the body P, cc&teris paribus, is more curve at the 
quadratures than at the conjunction and opposition. For the swifter 
bodies move, the less they deflect from a rectilinear path. And besides the 
force KL, or NM, at the conjunction and opposition, is contrary to the 
force with which the body T attracts the body P, and therefore diminishes 
that force ; but the body P will deflect the less from a rectilinear path the 
less it is impelled towards the body T. 

COR. 5. Hence the body P, cceteris paribus, goes farther from the body 
T at the quadratures than at the conjunction and opposition. This is said, 
E C_ L 




B 

however, supposing no regard had to the motion of eccentricity. For if 
the orbit of the body P be eccentrical, its eccentricity (as will be shewn 
presently by Cor. 9) will be greatest when the apsides are in the syzy- 
gies; and thence it may sometimes come to pass that the body P. in its 
near approach to the farther apsis, may go farther from the body T at the 
syzygies than at the quadratures. 

COR. 6. Because the centripetal force of the central body T, by which 



206 THE MATHEMATICAL PRINCIPLES [BOOK. 1 

the body P is retained in its orbit, is increased at the quadratures by tho 
addition caused by the force LM, and diminished at the syzygies by the 
subduction caused by the force KL, and, because the force KL is greater 
than LM, it is more diminished than increased ; and, moreover, since that 
centripetal force (by Cor. 2, Prop. IV) is in a ratio compounded of the sim 
ple ratio of the radius TP directly, and the duplicate ratio of the periodi 
cal time inversely ; it is plain that this compounded ratio is diminished by 
the action of the force KL ; and therefore that the periodical time, supposing 
the radius of the orbit PT to remain the same, will be increased, and that 
in the subduplicate of that ratio in which the centripetal force is diminish 
ed ; and, therefore, supposing this radius increased or diminished, the peri 
odical time will be increased more or diminished less than in the sesquipli- 
cate ratio of this radius, by Cor. 6, Prop. IV. If that force of the central 
body should gradually decay, the body P being less and less attracted would 
go farther and farther from the centre T ; and, on the contrary, if it were 
increased, it would draw nearer to it. Therefore if the action of the distant 
body S, by which that force is diminished, were to increase and decrease 
by turns, the radius TP will be also increased and diminshed by turns ; 
and the periodical time will be increased and diminished in a ratio com 
pounded of the sesquiplicate ratio of the radius, and of the subduplicate oi 
that ratio in which the centripetal force of the central body T is dimin 
ished or increased, by the increase or decrease of the action of the distant 
body S. 

COR. 7. It also follows, from what was before laid down, that the axis 
of the ellipsis described by the body P, or the line of the apsides, does as 
to its angular motion go forwards and backwards by turns, but more for 
wards than backwards, and by the excess of its direct motion is in the 
whole carried forwards. For the force with which the body P is urged to 
the body T at the quadratures, where the force MN vanishes, is compound 
ed of the force LM and the centripetal force with which the body T at 
tracts the body P. The first force LM, if the distance PT be increased, is 
increased in nearly the same proportion with that distance, and the other 
force decreases in the duplicate ratio of the distance ; and therefore the 
sum of these two forces decreases in a less than the duplicate ratio of the 
distance PT ; and therefore, by Cor. 1, Prop. XLV, will make the line of 
the apsides, or, which is the same thing, the upper apsis, to go backward. 
But at the conjunction and opposition the force with which the body P is 
urged towards the body T is the difference of the force KL, and of the 
force with which the body T attracts the body P ; and that difference, be 
cause the force KL is very nearly increased in the ratio of the distance 
PT, decreases in more -than the duplicate ratio of the distance PT ; and 
therefore, by Cor. 1, Prop. XLV, causes the line of the apsides to go for 
wards. In the places between the syzygies and the quadratures, the motion 



SEC. Xl.J OF NATURAL PHILOSOPHY. 207 

of the line of the apsides depends upon both < f these causes conjuncdy, so 
that it either goes forwards or backwards in proportion to the excess ol 
one of these causes above the other. Therefore since the force KL in the 
syzygies is almost twice as great as the force LM in the quadratures, the 
excess will be on the side of the force KL, and by consequence the line of 
the apsides will be carried forwards. The truth of this arid the foregoing 




IE 

Corollary will be more easily understood by conceiving the system of the 
two bodies T and P to be surrounded on every side by several bodies S, 
S, S, dec., disposed about the orbit ESE. For by the actions of these bo 
dies the action of the body T will be diminished on every side, and decrease 
in more than a duplicate ratio of the distance. 

COR. 8. IJut since the progress or regress of the apsides depends upon 
the decrease of the centripetal force, that is, upon its being in a greater or 
less ratio than the duplicate ratio of the distance TP, in the passage of 
the body from the lower apsis to the upper ; and upon a like increase in 
its return to the lower apsis again ; and therefore becomes greatest where 
the proportion of the force at the upper apsis to the force at the lower ap 
sis recedes farthest from the duplicate ratio of the distances inversely ; it 
is plain, that, when the apsides are in the syzygies, they will, by reason of 
the subducting force KL or NM LM, go forward more swiftly ; and in 
the quadratures by the additional force LM go backward more slowly. 
Because the velocity of the progress or slowness of the regress is continued 
for a long time ; this inequality becomes exceedingly great. 

COR. 9. If a body is obliged, by a force reciprocally proportional to the 
square of its distance from any centre, to revolve in an ellipsis round that 
centre ; and afterwards in its descent from the upper apsis to the lower 
apsis, that force by a perpetual accession of new force is increased in more 
than a duplicate ratio of the diminished distance ; it is manifest that the 
body, being impelled always towards the centre by the perpetual accession 
of this new force, will incline more towards that centre than if it were 
urged by that force alone which decreases in a duplicate ratio of the di 
minished distance, and therefore will describe an orbit interior to that 
elliptical orbit, and at the lower apsis approaching nearer to the centre 
than before. Therefore the orbit by the accession of this new force will 
become more eccentrical. If now, while the body is returning from the 
lower to the upper apsis, it should decrease by the same degrees by which 
it increases before the body would return to its first distance; and there- 



THE MATHEMATICAL PRINCIPLES [BOOK I. 

fore if the force decreases in a yet greater ratio, the body, being now less 
attracted than before, will ascend to a still greater distance, and so the ec 
centricity of the orbit will be increased still more. Therefore if the ratio 
of the increase and decrease of the centripetal force be augmented each 
revolution, the eccentricity will be augmented also ; and, on the contrary, 
if that ratio decrease, it will be diminished. 

Now, therefore, in the system of the bodies T, P, S, when the apsides of 
the orbit FAB are in the quadratures, the ratio of that increase and de 
crease is least of all, and becomes greatest when the apsides are in the 
syzygies. If the apsides are placed in the quadratures, the ratio near the 
apsides is less, and near the syzygies greater, than the duplicate ratio of the 
distances : and from that Greater ratio arises a direct motion of the line of 

7 o 

the apsides, as was just now said. But if we consider the ratio of the 
whole increase or decrease in the progress between the apsides, this is less 
than the duplicate ratio of the distances. The force in the lower is to the 
force in the upper apsis in less than a duplicate ratio of the distance of the 
upper apsis from the focus of the ellipsis to the distance of the lower apsis 
from the same focus ; and, contrariwise, when the apsides are placed in the 
syzygies, the force in the lower apsis is to the force in the upper apsis in a 
greater than a duplicate ratio of the distances. For the forces LM in the 
quadratures added to the forces of the body T compose forces in a less ra 
tio ; and the forces KL in the syzygies subducted from the forces of the 
body T, leave the forces in a greater ratio. Therefore the ratio of the 
whole increase and decrease in the passage between the apsides is least at 
the quadratures and greatest at the syzygies ; and therefore in the passage 
of the apsides from the quadratures to the syzygies it is continually aug 
mented, and increases the eccentricity of the ellipsis ; and in the passage 
from the syzygies to the quadratures it is perpetually decreasing, and di 
minishes the eccentricity. 

COR. 10. That we may give an account of the errors as to latitude, let 
us suppose the plane of the orbit EST to remain immovable; and from 
the cause of the errors above explained, it is manifest, that, of the two 
forces NM, ML, which are the only and entire cause of them, the force 
ML acting always in the plane of the orbit PAB never disturbs the mo 
tions as to latitude ; and that the force NM, when the nodes are in the 
gyzygies, acting also in the same plane of the orbit, does not at that time 
affect those motions. But when the nodes are in the quadratures, it dis 
turbs tliem very much, and, attracting the body P perpetually out of the 
plane of its orbit, it diminishes the inclination of the plane in the passage 
of the body from the quadratures to the syzygies, and again increases the 
same in the passage from the syzygies to the quadratures. Hence it 
comes to pass that when the body is in the syzygies, the inclination is 
then least of all, and returns to the first magnitude nearly, when the body 



SEC. XL] OF NATURAL PHILOSOPHY. 209 

arrives at the next node. But if the nodes are situate at the octants after 
the quadratures, that is, between C and A, D and B, it will appear, from 

ii C L 




E 

wnat was just now shewn, that in the passage of the body P from either 
node to the ninetieth degree from thence, the inclination of the plane is 
perpetually diminished ; then, in the passage through the next 45 degrees 
to the next quadrature, the inclination is increased ; and afterwards, again, 
in its passage through another 45 degrees to the next node, it is dimin 
ished. Therefore the inclination is more diminished than increased, and 
is therefore always less in the subsequent node than in the preceding one. 
And, by a like reasoning, the inclination is more increased than diminish 
ed when the nodes are in the other octants between A and D, B and C. 
The inclination, therefore, is the greatest of all when the nodes are in the 
syzygies In their passage from the syzygies to the quadratures the incli 
nation is diminished at each appulse of the body to the nodes : and be 
comes least of all when the nodes are in the quadratures, and the body in 
the syzygies ; then it increases by the same degrees by which it decreased 
before ; and, when the nodes come to the next syzygies, returns to its 
former magnitude. 

COR. 11. Because when the nodes are in the quadratures the body P is 
perpetually attracted from the plane of its orbit ; and because this attrac 
tion is made towards S in its passage from the node C through the con 
junction A to the node D ; and to the contrary part in its passage from the 
node D through the opposition B to the node C; it is manifest that, in its 
motion from the node C, the body recedes continually from the former 
plane CD of its orbit till it comes to the next node; and therefore at that 
node, being now at its greatest distance from the first plane CD, it will 
pass through the plane of the orbit EST not in D, the other node of that 
plane, but in a point that lies nearer to the body S, which therefore be 
comes a new place of the node in, antecedent ia to its former place. And, 
by a like reasoning, the nodes will continue to recede in their passage 
from this node to the next. The nodes, therefore, when situate in the 
quadratures, recede perpetually ; and at the syzygies, where no perturba 
tion can be produced in the motion as to latitude, are quiescent : in the in 
termediate places they partake of both conditions, and recede more slowly ; 
and, therefore, being always either retrograde or stationary, they will be 
carried backwards, or in atitecedentia, each revolution. 

COR. 12. All the errors described in these corrollaries arc a little greater 

14 



210 THE MATHEMATICAL PRINCIPLES BOOK L 

at the conjunction of the bodies P, S, than at their opposition ; because 
the generating forces NM and ML are greater. 

COR. 13. And since the causes and proportions of the errors and varia 
tions mentioned in these Corollaries do not depend upon the magnitude of 
the body S, it follows that all things before demonstrated will happen, if 
the magnitude of the body S be imagined so great as that the system of the 
two bodies P and T may revolve about it. And from this increase of the 
body S, and the consequent increase of its centripetal force, from which the 
errors of the body P arise, it will follow that all these errors, at equal dis 
tances, will be greater in this case, than in the other where the body S re 
volves about the system of the bodies P and T. 

COR. 14. But since the forces NM, ML, when the body S is exceedingly 
distant, are very nearly as the force SK and the ratio PT to ST con- 
junctly ; that is, if both the distance PT, and the absolute force of the body 
8 be given, as ST 3 reciprocally : and since those forces NM, ML are the 
causes of all the errors and effects treated of in the foregoing Corollaries; 
it is manifest that all those effects, if the system of bodies T and P con 
tinue as before, and only the distance ST and the absolute force of the body 
S be changed, will be very nearly in a ratio compounded of the direct ratio 
of the absolute force of the body S, and the triplicate inverse ratio of the 
distance ST. Hence if the system of bodies T and P revolve about a dis 
tant body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and 
6, Prop IV) reciprocally in a duplicate ratio c/f the periodical time. And 
thence, also, if the magnitude of the bodv S be proportional to its absolute 
force, those forces NM, ML, and their effects, will be directly as the cube 
of the apparent diameter of the distant body S viewed from T, and so vice 
versa. For these ratios are the same as the compounded ratio above men 
tioned. 

COR. 15. And because if the orbits ESE and PAB, retaining their fig 
ure, proportions, and inclination to each other, should alter their magni 
tude ; arid the forces of the bodies S and T should either remain, or be 
changed in any given ratio ; these forces (that is, the force of the body T, 
which obliges the body P to deflect from a rectilinear course into the orbit 
PAB, and the force of the body S, which causes the body P to deviate from 
that orbit) would act always in the same manner, and in the same propor 
tion : it follows, that all the effects will be similar and proportional, arid 
the times of those effects proportional also ; that is, that all the linear er 
rors will be as tne diameters of the orbits, the angular errors the same as 
before ; and the times of similar linear errors, or equal angular errors ? as 
the periodical times of the orbits. 

COR. 16. Therefore if the figures of the orbits and their inclination to 
each other be given, and the magnitudes, forces, arid distances of the bodies 
he any how changed, we may. from the errors and times of those errors in 



SEC. XI.] OF NATURAL PHILOSOPHY. 2 \\ 

one case, collect very nearly the errors and times of the errors in any other 
case. But this may be done more expeditiously by the following method. 
The forces NM ; ML, other things remaining unaltered, are as the radius 
TP ; and their periodical effects (by Cor. 2, Lein. X) are as the forces and 
the square of the periodical time of the body P conjunctly. These are the 
linear errors of the body P ; and hence the angular errors as they appear 
from the centre T (that is, the motion of the apsides and of the nodes, and all 
the apparent errors as to longitude and latitude) are in each revolution of 
the body P as the square of the time of the revolution, very nearly. Let 
these ratios be compounded with the ratios in Cor. 14, and in any system 
of bodies T, P, S, where P revolves about T very near to it, and T re 
volves about S at a great distance, the angular errors of the body P, ob 
served from the centre T, will be in each revolution of the body P as the 
square of the periodical time of the body P directly, and the square of the 
periodical time of the body T inversely. And therefore the mean motion 
of the line of the apsides will be in a given ratio to the mean motion of 
the nodes ; and both those motions will be as the periodical time of the 
body P directly, and the square of the periodical time of the body T in 
versely. The increase or diminution of the eccentricity and inclination of 
the orbit PAB makes no sensible variation in the motions of the apsides* 
and nodes, unless that inc/case or diminution be very great indeed. 

COR. 17. Sines the line LM becomes sometimes greater and sometimes 
less than the radius PT, let the mean quantity of the force LM be expressed 
E C 



sa - -::-..::::::; 




by that radius PT ; and then that mean force will be to the mean force 
SK or SN (which may be also expressed by ST) as the length PT to the 
length ST. But the mean force SN or ST, by which the body T is re 
tained in the orbit it describes about S, is to the force with which the body P 
is retained in its orbit about T in a ratio compounded of the ratio of the 
radius ST to the radius PT, and the duplicate ratio of the periodical time 
of the body P about T to the periodical time of the body T about S. And, 
ex cequo, the mean force LM is to the force by which the body P is retain 
ed in its orbit about T (or by which the same body P might revolve at the 
distance PT in the same periodical time about any immovable point T) in 
the same duplicate ratio of the periodical times. The periodical times 
therefore being given, together with the distance PT, the mean force LM 
is also given ; and that force being given, there is given also the force MN, 
very nearly, by the analogy of the lines PT and MN. 



212 THE MATHEMATICAL PRINCIPLES [BoOK I 

Con. IS. By tlie same laws by which the body P revolves about the 
body T, let us suppose many fluid bodies to move round T at equal dis 
tances from it ; and to be so numerous, that they may all become contiguous 
to each other, so as to form a fluid annul us, or ring, of a round figure, and 
concentrical to the body T; and the several parts of this annulus, perform 
ing their motions by the same law as the body P, will draw nearer to the 
body T, and move swifter in the conjunction and opposition of themselves 
and the body S, than in the quadratures. And the nodes of this annulus, 
or its intersections with the plane of the orbit of the body S or T, will rest 
at the syzygies ; but out of the syzygies they will be carried backward, or 
in. antecedentia ; with the greatest swiftness in the quadratures, and more 
slowly in other places. The inclination of this annulus also will vary, and 
its axis will oscillate each revolution, and when the revolution is completed 
will return to its former situation, except only that it will be carried round 
a little by the precession of the nodes. 

COR. 19. Suppose now the spherical body T, consisting of some matter 
not fluid, to be enlarged, and to extend its If on every side as far as that 
annulus, and that a channel were cut all round its circumference contain 
ing water j and that this sphere revolves uniformly about its own axis in 
the same periodical time. This water being accelerated and retarded by 
turns (as in the last Corollary), will be swifter at the syzygies, and slower 
at the quadratures, than the surface of the globe, and so will ebb and flow in 
its channel after the manner of the sea. If the attraction of the body S were 
taken away, the water would acquire no motion of flux and reflux by revolv- 
.ng round the quiescent centre of the globe. The case is the same of a globe 
moving uniformly forwards in a right line, and in the mean time revolving 
about its centre (by Cor. 5 of the Laws of Motion), and of a globe uni 
formly attracted from its rectilinear course (by Cor. 6, of the same Laws). 
But let the body S come to act upon it, and by its unequable attraction the 
A\ater will receive this new motion ; for there will be a stronger attraction 
upon that part of the water that is nearest to the body, and a weaker upon 
that part which is more remote. And the force LM will attract the water 
downwards at the quadratures, and depress it as far as the syzygies ; and the 
force KL will attract it upwards in the syzygies, and withhold its descent, 
and make it rise as far as the quadratures ; except only in so far as the 
motion of flux and reflux may be directed by the channel of the water, and 
be a little retarded by friction. 

COR. 20. If, now, the annulus becomes hard, and the globe is diminished, 
the motion of flux and reflux will cease ; but the oscillating motion of the 
inclination and the praecession of the nodes will remain. Let the globe 
have the same axis with the annulus, and perform its revolutions in the 
same times, and at its surface touch the annulus within, and adhere to it; 
then the globe partaking of the motion of the annulus, this whole compares 



SEC. XI. OF NATURAL PHILOSOPHY. 213 

will oscillate, and the nodes will go backward, for the globe, as \ve shall 
shew presently, is perfectly indifferent to the receiving of all impressions. 
The greatest angle of the inclination of the annulus single is when the 
nodes are in the syzygies. Thence in the progress of the nodes to the 
quadratures, it endeavours to diminish its inclination, and by that endea 
vour impresses a motion upon the whole globe. The globe retains this 
motion impressed, till the annulus by a contrary endeavour destroys that 
motion, and impresses a new motion in a contrary direction. And by this 
means the greatest motion of the decreasing inclination happens when the 
nodes are in the quadratures; and the least angle of inclination in the octants 




B 

after the quadratures ; and, again, the greatest motion of roclination happens 
when the nodes are in the syzygies ; and the greatest angle of reclination in 
the octants following. And the case is the same of a globe without this an 
nulus, if it be a little higher or a little denser in the equatorial than in the 
polar regions : for the excess of that matter in the regions near the equator 
supplies the place of the annulus. And though we should suppose the cen 
tripetal force of this globe to be any how increased, so that all its parts 
were to tend downwards, as the parts of our earth gravitate to the centre, 
yet the phenomena of this and the preceding Corollary would scarce be al 
tered ; except that the places of the greatest and least height of the water 
will be different : for the water is now no longer sustained and kept in its 
orbit by its centrifugal force, but by the channel in which it flows. And, 
besides, the force LM attracts the water downwards most in the quadra 
tures, and the force KL or NM LM attracts it upwards most in the 
syzygies. And these forces conjoined cease to attract the water downwards, 
and begin to attract it upwards in the octants before the syzygies ; and 
cease to attract the water upwards, and begin to attract the water down 
wards in the octants after the syzygies. And thence the greatest height of 
the water may happen about the octants after the syzygies ; and the least 
height about the octants after the quadratures ; excepting only so far as the 
motion of ascent or descent impressed by these forces may by the vis insita 
of the water continue a little longer, or be stopped a little sooner by impe 
diments in its channel. 

COR. 21. For the same reason that redundant matter in the equatorial 
regions of a globe causes the nodes to go backwards, and therefore by the 
increase of that matter that retrogradation is increased, by the diminution 
is diminished, and by the removal quite ceases : it follows, that, if more than 



214 THE MATHEMATICAL PRINCIPLES [BOOK I 

that redundant matter be taken away, that is, if the globe be either more 
depressed, or of a more rare consistence near the equator than near the 
poles, there will arise a motion of the nodes in consequentia. 

COR. 22. And thence from the motion of the nodes is known the consti 
tution of the globe. That is, if the globe retains unalterably the same poles, 
and the motion (of the nodes) be in. antecedetitia, there is a redundance oi 
the matter near the equator; but if in conseqnentia, a deficiency. Sup 
pose a uniform and exactly spherical globe to be first at rest in a free space : 
then by some impulse made obliquely upon its superficies to be driven from 
its place, and to receive a motion partly circular and partly right forward. 
Because this globe is perfectly indifferent to all the axes that pass through 
its centre, nor has a greater propensity to one axis or to one situation oi 
the axis than to any other, it is manifest that by its own force it will never 
change its axis, or the inclination of it. Let now this globe be impelled 
obliquely by a new impulse in the same part of its superficies as before . 
and since the effect of an impulse is not at all changed by its coming sooner 
or later, it is manifest that these two impulses, successively impressed, will 
produce the same motion as if they were impressed at the same time : that 
is, the same motion as if the globe had been impelled by a simple force 
compounded of them both (by Cor. 2, of the Laws), that is, a simple motion 
about an axis of a given inclination. And the case is the same if the sec 
ond impulse were made upon any other place of the equator of the first 
motion ; and also if the first impulse were made upon any place in the 
equator of the motion which would be generated by the second impulse 
alone; and therefore, also, when both impulses are made in any places 
whatsoever ; for these impulses will generate the same circular motion as 
if they were impressed together, and at once, in the place of the intersec 
tions of the equators of those motions, which would be generated by each 
of them separately. Therefore, a homogeneous and perfect globe will not 
retain several distinct motions, but will unite all those that are impressed 
on it, and reduce them into one; revolving, as far as in it lies, always with 
a simple and uniform motion about one single given axis, with an inclina 
tion perpetually invariable. And the inclination of the axis, or the velocity 
of the rotation, will not be changed by centripetal force. For if the globe 
be supposed to be divided into two hemispheres, by any plane whatsoever 
passing through its own centre, and the centre to which the force is direct 
ed, that force will always urge each hemisphere equally ; and therefore will 
not incline the globe any way as to its motion round its own axis. But 
let there be added any where between the pole and the equator a heap oi 
new matter like a mountain, and this, by its perpetual endeavour to recede 
from the centre of its motion, will disturb the motion of the globe, and 
cause its poles to wander about its superficies, describing circles about 
themselves and their opposite points. Neither can this enormous evagatior 



XL] OF NATURAL PHILOSOPHY. 2 In 

of the poles be corrected, unless by placing that mountain ei . er in one ol 
the poles; in which case, by Cor. 21, the nodes of the equator will go for 
wards ; or in the equatorial regions, in which case, by Cor. 20, the nodes 
will go backwards: or, lastly, by adding on the other side of the axis anew 
quantity of matter, by which the mountain may be balanced in its motion; 
and then the nodes will either go forwards or backwards, as the mountain 
and this newly added matter happen to be nearer to the pole or to the 
equator. 

PROPOSITION LXV1I. THEOREM XXVII. 

The same laics of attraction being supposed, I say, that the exterior body 
S does, by radii dra.cn to the point O, the common centre of gravity 
of the interior bodies P and T, describe round that centre areas more 
proportional to the times, and an orbit more approaching to the form 
of an ellipsis having its focus in that cen > .-. than, it can describe 
round the innermost and greatest body T by ra Hi drawn to that 
body. 
For the attractions of the body S towards T and 

P compose its absolute attraction, which is more 

directed towards O, the common centre of gravity S(i 

of the bodies T and P, than it is to the . reatest 

body T ; and which is more in a reciprocal propor 
tion to the square of the distance SO, than it is to the square of the distance 

ST : as will easily appear by a little consideration. 

PROPOSITION LXVIII. THEOREM XXVIII. 

The same laws of attraction supposed, I say, that the exterior body S 
will, by radii drawn to O, the common centre of gravity of the interior 
bodies P and T, describe round that centre areas more propor 
tional to the times, and an orbit more approaching to the form of an 
ellipsis having its focus in that centre, if the innermost and greatest 
body be agitated by these attractions as well as the rest, than it would 
do if that body were either at rest as not attracted, or were much tnore 
or much less attracted, or much more or much less agitated. 
This may be demonstrated after the same manner as Prop. LXVI, but 
by a more prolix reasoning, which I therefore pass over. It will be suf 
ficient to consider it after this manner. From the demonstration of the 
last Proposition it is plain, that the centre, towards which the body S is 
urged by the two forces conjunctly, is very near to the common centre of 
gravity of those two other bodies. If this centre were to coincide with that 
common centre, and moreover the common centre of gravity of all the three 
bodies were at rest, the body S on one side, and the common centre of 
gravity of the other two bodies on the other side, would describe true ellip* 





216 THE MATHEMATICAL PRINCIPLES - [BOOK 1 

ses about that quiescent common centre. This appears from Cor. 2, Pro]) 
LVIII, compared with what was demonstrated in Prop. LX1V, and LXY 
Now this accurate elliptical motion will be disturbed a little by the dis 
tance of the centre of the two bodies from the centre towards which tht 
third body S is attracted. Let there be added, moreover, a motion to the 
Bommon centre of the three, and the perturbation will be increased yet 
more. Therefore the perturbation is least when the 
common centre of the three bodies is at rest; that 
I is, when the innermost and greatest body T is at 
tracted according to the same law as the rest are ; 
and is always greatest when the common centre of 
the three, by the diminution of the motion of the body T, begins to be 
moved, and is more and more agitated. 

COR. And hence if more lesser bodies revolve about the great one, it 
may easily be inferred that the orbits described will approach nearer to 
ellipses ; and the descriptions of areas will be more nearly equable, if all 
the bodies mutually attract and agitate each other with accelerative forces 
that are as their absolute forces directly, and the squares of the distances 
inversely : and if the focus of each orbit be placed in the common centre 
of gravity of all the interior bodies (that is. if the focus of the first and in 
nermost orbit be placed in the centre of gravity of the greatest and inner 
most body : the focus of the second orbit in the common centre of gravity 
of the two innermost bodies; the focus of the third orbit in the common 
centre of gravity of the three innermost ; and so on), than if the innermost 
body were at rest, and was made the common focus of all the orbits. 

PROPOSITION LXIX. THEOREM XXIX. 

fn a system of several bodies A, B, C, D, $*c., if any one of those bodies, 
as A, attract all the rest, B, C, D, $*c.,with accelerative forces that are 
reciprocally as the squares of the distances from the attracting body ; 
and another body, as B, attracts also the rest. A, C, D, $-c., with forces 
that are reciprocally as the squares of the distances from the attract 
ing body ; the absolute forces of the attracting bodies A and B will 
be to each other as those very bodies A and B to which those forces 
belong. 

For the accelerative attractions of all the bodies B, C, D, towards A, 
are by the supposition equal to each other at equal distances ; and in like 
manner the accelerative attractions of all the bodies towards B are also 
equal to each other at equal distances. But the absolute attractive force 
of the body A is to the absolute attractive force of the body B as the ac- 
eelerative attraction of all the bodies towards A to the accelerative attrac 
tion of all the bodies towards B at equal distances ; and so is also the ac 
celerative attraction of the body B to*vards A to the accelerative attraction 



SEC. XI] OF NATURAL PHILOSOPHY. 21 T 

of the body A towards B. But the accelerative attraction of the body B 
towards A is to the accelerative attraction of the body A towards B as the 
mass of the body A to the mass of the body B ; because the motive forces 
which (by the 2d, 7th, and 8th Definition) are as the accelerative forces 
and the bodies attracted conjunctly are here equal to one another by the 
third Law. Therefore the absolute attractive force of the body A is to the 
absolute attractive force of the body B aa the mass of the body A to the 
mass of the body B. Q.E.D. 

COR. 1. Therefore if each of the bodies of the system A, B, C, D, &c. 
does singly attract all the rest with accelerative forces that are reciprocally 
as the squares of the distances from the attracting body, the absolute forces 
of all those bodies will be to each other as the bodies themselves. 

COR. 2. By a like reasoning, if each of the bodies of the system A, B, 
C, D, &c., do singly attract all the rest with accelerative forces, which are 
either reciprocally or directly in the ratio of any power whatever of the 
distances from the attracting body : or which are defined by the distances 
from each of the attracting bodies according to any common law : it is plain 
that the absolute forces of those bodies are as the bodies themselves. 

COR. 3. In a system of bodies whose forces decrease in the duplicate ra 
tio of the distances, if the lesser revolve about one very great one in ellip 
ses, having their common focus in the centre of that great body, and of a 
figure exceedingly accurate ; and moreover by radii drawn to that great 
ody describe areas proportional to the times exactly the absolute forces 
)i those bodies to each other will be either accurately or very nearly in the 
ratio of the bodies. And s > on the contrary. This appears from Cor. of 
Prop. XLVII1, compared with the first Corollary of this Prop. 

SCHOLIUM. 

These Propositions naturally lead us to the analogy there is between 
centripetal forces, and the central bodies to which those forces used to be 
directed ; for it is reasonable to suppose that forces which are directed to 
bodies should depend upon the nature and quantity of those bodies, as we 
see they do in magnetical experiments. And when such cases occur, we 
are to compute the attractions of the bodies by assigning to each of their 
particles its proper force, and then collecting the sum of them all. I here 
ue*e the word attraction in general for any endeavour, of what kind soever, 
made by bodies to approach to each other; whether that endeavour arise 
from the action of the bodies themselves, as tending mutually to or agita 
ting each other by spirits emitted; or whether it arises from the action 
of the aether or of the air, or of any medium whatsoever* whether corporeal 
or incorporeal, any how impelling bodies placed therein towards each other. 
In the same general sense I use the word impulse, not defining in this trea 
tise the species or physical qualities of forces, but investigating the quantities 



THE MATHEMATICAL PRINCIPLES [BOOK ). 

and mathematical proportions of them ; as I observed before ir (lie Defi 
nitions. In mathematics we are to investigate the quantities of forces 
with their proportions consequent upon any conditions supposed ; then, 
when we enter upon physics, we compare those proportions with the phe 
nomena of Nature, that we may know what conditions of those forces an 
swer to the several kinds of attractive bodies. And this preparation being 
made, we argue more safely concerning the physical species, causes, and 
proportions of the forces. Let us see, then, with what forces spherical 
bodies consisting of particles endued with attractive powers in the manner 
above spoken of must act mutually upon one another : and what kind of 
motions will follow from thence. 




SECTION XII. 

Of the attractive forces of sphcerical bodies. 

PROPOSITION LXX. THEOREM XXX. 

If to every point of a spherical surface there tend equal centripetal forces 
decreasing in, the duplicate ratio of the distances from those points ; 
I say, that a corpuscle placed within that superficies will not be attract 
ed by those forces any way. 

Let HIKL, be that sphaerical superficies, and P a 
corpuscle placed within. Through P let there be 
drawn to this superficies to two lines HK, IL, inter- 
cepting very small arcs HI, KL ; and because (by 
Cor. 3, Lem. VII) the triangles HPI,LPK are alike, 
those arcs will be proportional to the distances HP 
LP ; and any particles at HI and KL of the spheri 
cal superficies, terminated by right lines passing through P, will be in the 
duplicate ratio of those distances. Therefore the forces of these particles 
exerted upon the body P are equal between themselves. For the forces are 
as the particles directly, and the squares of the distances inversely. And 
these two ratios compose the ratio of equality. The attractions therefore, 
being made equally towards contrary parts, destroy each other. And by a 
like reasoning all the attractions through the whole spherical superficies 
are destroyed by contrary attractions. Therefore the body P will not be 
any way impelled by those attractions. Q.E.D. 

PROPOSITION LXXI. THEOREM XXXI. 

The same things supposed as above, I say, that a cor pu vie placed with 
out the sph(ericl superficies is attracted towards the centre of tht 
sphere wiih a force reciprocally proportional to the square of its dis 
tance from that centre. 
Let AHKB, ahkb, be two equal sphaerical superficies described about 



SEC. XII.J OF NATURAL PHILOSOPHY. 

the centre S, s ; their diameters AB, ab ; and let P and p be two corpus 
cles situate without the gpheres in those diameters produced. Let there 





be drawn from the corpuscles the lines PHK, PIL, phk, pil, cutting off 
from the great circles AHB, ahb, the equal arcs HK, hk, IL ; il ; and to 
those lines let fall the perpendiculars SD, sd, SE, SP, 1R, ir ; of which let 
SD, sd, cut PL, pi, in F and f. Let fall also to the diameters the perpen 
diculars IQ, iq. Let now the angles DPE, dpe, vanish; and because DS 
and ds, ES and es are equal, the lines PE, PP, and pe, pf, and the lineolso 
I )F, df may be taken for equal ; because their last ratio, when the angles 
DPE, dpe vanish together, is the ratio of equality. These things then 
supposed, it will be, as PI to PF so is RI to DF, and as pf to pi so is df or 
DF to ri ; and, ex cequo, as PI X pf to PF X pi so is R I to ri, that is 
(by Cor. 3, Lem VII), so is the arc IH to the arc ih. Again, PI is to PS 
as IQ. to SE, and ps to pi as se or SE to iq ; and, ex ceqno, PI X ps to 
PS X pi as IQ. to iq. And compounding the ratios PI 2 X pf X ps is to 
pi 2 X PF X PS, as IH X IQ to ih X iq ; that is, as the circular super 
ficies which is described by the arc IH, as the semi-circle AKB revolves 
about the diameter AB, is to the circular superficies described by the arc ih 
as the semi-circle akb revolves about the diameter ab. And the forces 
with which these superficies attract the corpuscles P and p in the direction 
of lines tending to those superficies are by the hypothesis as the superficies 
themselves directly, and the squares of the distances of the superficies from 
those corpuscles inversely; that is, as pf X ps to PF XPS. And these 
forces again are to the oblique parts of them which (by the resolution of 
forces as in Cor. 2, of the Laws) tend to the centres in the directions of the 
lines PS, JDS-, as PI to PQ, and pi to pq ; that is (because of the like trian 
gles PIQ and PSF, piq and psf\ as PS to PF and ps to pf. Thence ex 
cequO) the attraction of the corpuscle P towards S is to the attraction of 

PF XpfXps. pf X PF X PS . 
the corpusclejo towards 5 as ~ = is to , that is, 

as ps 2 to PS 2 . And, by a like reasoning, the forces with which the su 
perficies described by the revolution of the arcs KL, kl attract those cor 
puscles, will be as jDS 2 to PS 2 . And in the same ratio will be the foroes 
of all the circular superficies into which each of the sphaerical superficies 
may be divided by taking sd always equal to SD, and se equal to SE. And 
therefore, by composition, the forces of the entire spherical superficies ex 
erted upon those corpuscles will be in the same ratio. Q.E.D 



220 THE MATHEMATICAL PRINCIPLES [BOOK 1 

PROPOSITION LXXIL THEOREM XXXII. 

If to the several points of a sphere there tend equal centripetal forces de 
creasing in a duplicate ratio of the distances from those points ; and 
there be given both the density of the sphere and the ratio of the di 
ameter of the sphere to the distance of the corpuscle from its centre ; 
I say, that the force with which the corpuscle is attracted is propor 
tional to the semi-diameter of the sphere. 

For conceive two corpuscles to be severally attracted by two spheres, one 
by one, the other by the other, and their distances from the centres of the 
spheres to be proportional to the diameters of the spheres respectively , and 
the spheres to be resolved into like particles, disposed in a like situation 
to the corpuscles. Then the attractions of one corpuscle towards the sev 
eral particles of one sphere will be to the attractions of the other towards 
as many analogous particles of the other sphere in a ratio compounded of 
the ratio of the particles directly, and the duplicate ratio of the distances 
inversely. But the particles are as the spheres, that is, in a triplicate ra 
tio of the diameters, and the distances are as the diameters ; and the first 
ratio directly with the last ratio taken twice inversely, becomes the ratio 
of diameter to diameter. Q.E.D. 

COR. 1. Hence if corpuscles revolve in circles about spheres composed 
of matter equally attracting, and the distances from the centres of the 
spheres be proportional to their diameters, the periodic times will be equal. 
COR. 2. And, vice versa, if the periodic times are equal, the distances 
will be proportional to the diameters. These two Corollaries appear from 
Cor. 3, Prop. IV. 

COR. 3. If to the several points of an^ two solids whatever, of like fig- 
are and equal density, there tend equal centripetal forces decreasing in a 
duplicate ratio of the distances from those points, the forces, with which 
corpuscles placed in a like situation to those two solids will be attracted 
by them, will be to each other as the diameters of the solids. 

PROPOSITION LXXIII. THEOREM XXXIII. 

If to the several points of a given sphere there tend equal centripetal forces 
decreasing in a duplicate ratio of the distances from the points ; 1 
say, that a corpuscle placed within the sphere is attracted by a force 
proportional to its distance from the centre. 

In the sphere ABCD, described about the centre S, 
let there be placed the corpuscle P ; and about the 
same centre S, with the interval SP ? conceive de- 
| B scribed an interior sphere PEQP. It is plain (by 
Prop. LXX) that the concentric sphaerical superficies, 
of which the difference AEBF of the spheres is com 
posed, have no effect at all upon the body P, their at- 




SEC. XIL] OF NATURAL PHILOSOPHY. 22\ 

tractions being destroyed by contrary attractions. There remains, there 
fore; only the attraction of the interior sphere PEQ,F. And (by Prop. 
LXXII) this is as the distance PS. Q.E.D. 

SCHOLIUM. 

By the superficies of which I here imagine the solids composed, I do not 
mean superficies purely mathematical, but orbs so extremely thin, that 
their thickness is as nothing; that is, the evanescent orbs of which the sphere 
will at last consist when the number of the orbs is increased, and their 
thickness diminished without end. In like manner, by the points of which 
lines, surfaces, and solids are said to be composed, are to be understood 
equal particles, whose magnitude is perfectly inconsiderable. 

PROPOSITION LXXIV. THEOREM XXXIV. 

The same things supposed, I say, that a corpuscle situate without the 

sphere is attracted with a force reciprocally proportional to the square 

of its distance from the centre. 

For suppose the sphere to be divided into innumerable concentric sphe 
rical superficies, and the attractions of the corpuscle arising from the sev 
eral superficies will be reciprocally proportional to the square of the dis 
tance of the corpuscle from the centre of the sphere (by Prop. LXXI). 
And, by composition, the sum of those attractions, that is, the attraction 
of the corpuscle towards the entire sphere, will be in the same ratio. Q.E.D. 

COR. 1. Hence the attractions of homogeneous spheres at equal distances 
from the centres will be as the spheres themselves. For (by Prop. LXXII) 
if the distances be proportional to the diameters of the spheres, the forces 
will be as the diameters. Let the greater distance be diminished in that 
ratio ; and the distances now being equal, the attraction will be increased 
in the duplicate of that ratio ; and therefore will be to the other attraction 
in the triplicate of that ratio ; that is, in the ratio of the spheres. 

COR. 2. At any distances whatever the attractions are as the spheres 
applied to the squares of the distances. 

COR. 3. If a corpuscle placed without an homogeneous sphere is attract 
ed by a force reciprocally proportional to the square of its distance from 
the centre, and the sphere consists of attractive particles, the force of ever y 
particle will decrease in a duplicate ratio of the distance from each particle. 

PROPOSITION LXXV. THEOREM XXXV. 

If to the several points of a given sphere there tend equal centripetal forces 
decreasing in a duplicate ratio of the distances from the points ; I say, 
that another similar sphere will be attracted by it with a force recip 
rocally proportional to the square of the distance of the centres. 
For the attraction of every particle is reciprocally as the square of its 



222 THE MATHEMATICAL PRINCIPLES | BOOK L 

distance from the centre of the attracting sphere (by Prop. LXXIV). and 
is therefore the same as if that whole attracting force issued from one sin 
gle corpuscle placed in the centre of this sphere. But this attraction is as 
great as on the other hand the attraction of the same corpuscle would be, 
if that were itself attracted by the several particles of the attracted sphere 
with the same force with which they are attracted by it. But that attrac 
tion of the corpuscle would be (by Prop. LXXIV) reciprocally propor 
tional to the square of its distance from the centre of the sphere : therefore 
the attraction of the sphere, equal thereto, is also in the same ratio. Q,.E.D. 

COR. 1. The attractions of spheres towards other homogeneous spheres 
are as the attracting spheres applied to the squares of the distances of their 
centres from the centres of those which they attract. 

COR. 2. The case is the same when the attracted sphere does also at 
tract. For the several points of the one attract the several points of the 
other with the same force with which they themselves are attracted by the 
others again; and therefore since in all attractions (by Law III) the at 
tracted and attracting point are both equally acted on, the force will be 
doubled by their mutual attractions, the proportions remaining. 

COR. 3. Those several truths demonstrated above concerning the motion 
of bodies about the focus of the conic sections will take place when an 
attracting sphere is placed in the focus, and the bodies move without the 
sphere. 

COR. 4. Those things which were demonstrated before of the motion of 
bodies about the centre of the conic sections take place when the motions 
are performed within the sphere. 

PROPOSITION LXXVI. THEOREM XXXVI. 

ff spheres be however dissimilar (as to density of matter and attractive, 
force] in the same ratio onward from the centre to the circumference ; 
but every where similar, at every given distance from the centre, on all 
sides round about ; and the attractive force of every point decreases 
in the duplicate ratio of the distance of the body attracted ; I say, 
that the whole force with which one of these spheres attracts the oilier 
will be reciprocally proportional to the square of the distance of the 
centres. 

Imagine several concentric similar 
spheres, AB, CD, EF, &c.. the inner 
most of which added to the outermost 
may compose a matter more dense to 
wards the centre, or subducted from 
them may leave the same more lax and 
rare. Then, by Prop. LXXV, these 
sphere? will attract other similar con- 




SEC. XII.] OF NATURAL PHILOSOPHY. 223 

eentric spheres GH ; IK, LM, &c., each the other, with forces reciprocally 
proportional to the square of the distance SP. And, by composition or 
division, the sum of all those forces, or the excess of any of them above 
the others; that is, the entire force with which the whole sphere AB (com 
posed of any concentric spheres or of their differences) will attract the 
whole sphere GH (composed of any concentric spheres or their differences) 
in the same ratio. Let the number of the concentric spheres be increased 
in infinitum, so that the density of the matter together with the attractive 
force may, in the progress from the circumference to the centre, increase or 
decrease according to any given law ; and by the addition of matter not at 
tractive, let the deficient density be supplied, that so the spheres may acquire 
any form desired ; and the force with which one of these attracts the other 
will be still, by the former reasoning, in the same ratio of the square of the 
distance inversely. Q.E.I). 

COR. I. Hence if many spheres of this kind, similar in all respects, at 
tract each other mutually, the accelerative attractions of each to each, at 
any equal distances of the centre s, will be as the attracting spheres. 

COR. 2. And at any unequal distances, as the attracting spheres applied 
to the squares of the distances between the centres. 

/ COR. 3. The motive attractions, or the weights of the spheres towards 
one another, will be at equal distances of the centres as the attracting and 
attracted spheres conjunctly ; that is, as the products arising from multi 
plying the spheres into each other. 

COR. 4. And at unequal distances, as those products directly, and the 
squares of the distances between the centres inversely. 

COR. 5. These proportions take place also when the attraction arises 
from the attractive virtue of both spheres mutually exerted upon each 
other. For the attraction is only doubled by the conjunction of the forces, 
the proportions remaining as before. 

COR. 6. If spheres of this kind revolve about others at rest, each about 
each ; and the distances between the centres of the quiescent and revolving 
bodies are proportional to the diameters of the quiescent bodies ; the peri 
odic times will be equal. 

COR. 7. And, again, if the periodic times are equal, the distances will 
be proportional to the diameters. 

COR. 8. All those truths above demonstrated, relating to the motions 
jf bodies about the foci of conic sections, will take place when an attract 
ing sphere, of any form and condition like that above described, is placed 
in the focus. 

COR. 9. And also when the revolving bodies are also attracting spheres 
Df any condition like that above described. 




224 THE MATHEMATICAL PRINCIPLES [BOOK I. 

PROPOSITION LXXVI1. THEOREM XXXVII. 

Tf to 1 he several points of spheres there tend centripetal forces propor 
tional to the distances of the points from the attracted bodies ; I say, 
that the compounded force with which two spheres attract each other 
mutually is as the distance between the centres of the spheres. 

CASE 1. Let AEBF be a sphere ; S its 
centre . P a corpuscle attracted : PA SB 
the axis of the sphere passing through the 
centre of the corpuscle ; EF, ef two planes 
cutting the sphere, and perpendicular to 
the axis, and equi-distant, one on one side, 
the other on the other, from the centre of 
the sphere ; G and g- the intersections of 
the planes and the axis ; and H any point in the plane EF. The centri 
petal force of the point H upon the corpuscle P, exerted in the direction of 
the line PH, is as the distance PH ; and (by Cor. 2, of the Laws) the same 
exerted in the direction of the line PG, or towards the . centre S, is as the 
length PG. Therefore the force of all the points in the plane EF (that is, 
of that whole plane) by which the corpuscle P is attracted towards the 
centre S is as the distance PG multiplied by the number of those points, 
that is, as the solid contained under that plane EF and the distance PG. 
And in like manner the force of the plane ef, by which the corpuscle P is 
attracted towards the centre S, is as that plane drawn into its distance Pg, 
or as the equal plane EF drawn into that distance Pg* ; and the sum of the 
forces of both planes as the plane EF drawn into the sum of the distances 
PG + P^, that is, as that plane drawn into twice the distance PS of the 
centre and the corpuscle ; that is, as twice the plane EF drawn into the dis 
tance PS, or as the sum of the equal planes EF + ef drawn into the same 
distance. And, by a like reasoning, the forces of all the planes in the 
whole sphere, equi-distant on each side from the centre of the sphere, are 
as the sum of those planes drawn into the distance PS, that is, as the 
whole sphere and the distance PS conjunctly. Q,.E.D. 

CASE 2. Let now the corpuscle P attract the sphere AEBF. And, by 
the same reasoning, it will appear that the force with which the sphere is 
attracted is as the distance PS. Q,.E.D. 

CASE 3. Imagine another sphere composed of innumerable corpuscles P : 
and because the force with which every corpuscle is attracted is as the dis 
tance of the corpuscle from the centre of the first sphere, and as the same 
sphere conjunctly, and is therefore the same as if it all proceeded from a 
single corpuscle situate in the centre of the sphere, the entire force with 
which all the corpuscles in the second sphere are attracted, that is, with 
which that whole sphere is attracted, will be the same as if that sphere 




SEC. Xll.] OP NATURAL PHILOSOPHY. 225 

were attracted by a force issuing from a single corpuscle in the centre of 
the first sphere ; and is therefore proportional to the distance between the 
centres of the spheres. Q,.E.D. 

CASE 4. Let the spheres attract each other mutually, and the force will 
be doubled, but the proportion will remain. Q..E.D. 

CASE 5. Let the corpuscle p be placed within ^- ^\E 

the sphere AEBF ; and because the force of the 
plane ef upon the corpuscle is as the solid contain 
ed under that plane and the distance jog ; and the 
contrary force of the plane EF as the solid con 
tained under that plane and the distance joG ; the ^ 
force compounded of both will be as the difference ** 
of the solids, that is, as the sum of the equal planes drawn into half the 
difference of the distances ; that is, as that sum drawn into joS, the distance 
of the corpuscle from the centre of the sphere. And, by a like reasoning, 
the attraction of all the planes EF, ef, throughout the whole sphere, that 
is, the attraction of the whole sphere, is conjunctly as the sum of all the 
planes, or as the whole sphere, and as joS, the distance of the corpuscle from 
the centre of the sphere. Q.E.D. 

CASE 6. And if there be composed a new sphere out of innumerable cor 
puscles such as jo, situate within the first sphere AEBF, it may be proved, 
as before, that the attraction, whether single of one sphere towards the 
other, or mutual of both towards each other, will be as the distance joS of 
the centres. Q, E.D. 

PROPOSITION LXXVIII. THEOREM XXXVIII. 

If spheres it* the progress from the centre to the circumference be hoivMtv 
dissimilar a->id unequable, but similar on every side round about af all 
given distances from the centre ; and the attractive force of evsrt/ 
point be as the distance of the attracted body ; I say, that the entire 
force with which two spheres of this kind attract each other mutitallij 
is proportional to the distance between the centres of the spheres. 
This is demonstrated from the foregoing Proposition, in the same man 
ner as Proposition LXXVI was demonstrated from Proposition LXXY. 

COR. Those things that were above demonstrated in Prop. X and LXJV, 
of the motion of bodies round the centres of conic sections, take place when 
all the attractions are made by the force of sphaerical bodies of the condi 
tion above described, and the attracted bodies are spheres of the same kind. 

SCHOLIUM. 

i have now explained the two principal cases of attractions; to wit, 
when the centripetal forces decrease in a duplicate ratio of the distances 
r increase in a simple ratio of the distances, causing the bodies in botli 

15 



226 THE MATHEMATICAL PRINCIPLES [BoOK 1 

cases to revolve in conic sections, and composing sphaerical bodies whose 
centripetal forces observe the same law of increase or decrease in the recess 
from the centre as the forces of the particles themselves do ; which is verv 
remarkable. It would be tedious to run over the other cases, whose con 
clusions are less elegant and important, so particularly as I have done 
these. I choose rather to comprehend and determine them all by one gen 
eral method as follows. 

LEMMA XXIX. 

ff about the centre S there be described any circle as AEB, and about the 
centre P there be. also described two circles EF, ef, cutting the Jirst in 
E and e, and the line PS in F and f ; and there be let fall to PS the 
perpendiculars ED, ed ; I say, that if the distance of the arcs EF ; ef 
be supposed to be infinitely diminished, the last ratio of the evanscent 
linr Dd to the evanescent line Ff is the same as that of the line PE to 
the live PS. 
For if the line Pe cut the arc EF in q ; and the right line Ee, which 




coincides with the evanescent arc Ee, be produced, and meet the right line 
PS in T ; and there be let fall from S to PE the perpendicular SG ; then, 
because of the like triangles DTE, </ !>, DES, it will be as Dd to Ee so 
))T to TE, or DE to ES : and because the triangles, Ee?, ESG (by Lem. 
VIII, and Cor. 3, Lem. VII) are similar, it will be as Ee to eq or F/soES 
to SG ; and, ex ceqno, as Dd to Ff so DE to SG ; that is (because of the 
similar triangles PDE ; PGS), so is PE to PS. Q.E.D. 

PROPOSITION LXXIX. THEOREM XXXIX. 

Suppose a superficies as EFfe to have its breadth infinitely diminished, 
and to be just vanishing ; and that the same superficies by its revolu- 
tion round the axis PS describes a spherical concavo-convex solid, to 
the several equnJ particle* of which there tend equal centripetal forces ; 
I soy, that the force with which thit solid attracts a corpuscle situate 
in P is in a ratio compounded of the ratio of the solid DE 2 X Ff and 
the ratio of the force with which the given particle in the place Ff 
would attract the same corpuscle. 
For if we consider, first, the force of the spherical superficies FE which 



SEC. xn.j 



OF NATURAL PHILOSOPHY. 



227 




is generated by the revolution of the arc FE, 
and is cut any where, as in r, by the line</6, 
the annular part of the super J cies generated 
by the revolution of the arc rE will be as the 
lineola Dd, the radius of the sphere PE re- 
mainiag the same; as Archimedes has de 
monstrated in his Book of the Sphere and 
Cylinder. And the force of this super 
ficies exerted in the direction of the lines PE 
or Pr situate all round in the conical superficies, will be as this annular 
superficies itself; that is as the lineola DC/, or, which is the same, as the 
rectangle under the given radius PE of the sphere and the lineola DC/ ; but 
that force, exerted in the direction of the line PS tending to the centre S, 
will be less in the ratio PI) to PE, and therefore will be as PD X DC/. 
Suppose now the line DF to be divided into innumerable little equal par 
ticles, each of which call DC/, and then the superficies FE will be divided 
into so many equal annuli, whose forces will be as the sum of all the rec 
tangles PD X DC/, that is, as |PF 2 - |PD 2 ; and therefore as DE-. 
Let now the superficies FE be drawn into the altitude F/; and the force 
of the solid EF/e exerted upon the corpuscle P will be as DE 2 X Ff; 
that is, if the force be given which any given particle as Ff exerts upon 
the corpuscle P at the distance PF. But if that force be not given, the 
force of the solid EF/e will be as the solid DE 2 X Ff and that force not 
given, conjunctly. Q.E.D. 

PROPOSITION LXXX. THEOREM XL. 

If to the several equal parts of a sphere ABE described about the centre 
S there tend equal centripetal forces ; and from the several points I) 
in the axis of the sphere AB in which a corpuscle, as F, is placed, 
there be erected the perpendiculars DE meeting the sphere in E, and 
if in those perpendiculars the lengths DN be taken as the quantity 

DE 2 X PS 

-, , and as th* force which a particle of the sphere situate in, 

the axis exerts at the distance PE upon the corpuscle P conjunctly ; ] 
say, that the in hole force with which the, corpuscle P is attracted to 
wards the sphere is as the area ANB, comprehended under the axis of 
the sphere AB, and the curve line ANB, the locus of the point N. 
For supposing the construction in the last Lemma and Theorem to 
stand, conceive the axis of the sphere AB to be divided into innumerable 
equal particles DC/, and the whole sphere to be divided into so many sphe 
rical concavo-convex laminae EF/e / and erect the perpendicular dn. By 
the last Theorem, the force with which the laminas EF/e attracts the cor 
puscle P is as DE 2 X Ff and the force of one particle exerted at the 



228 



THE MATHEMATICAL PRINCIPLES 



[BOOK I. 




distance PE or PF, conjunctly. 
But (by the last Lemma) Dd is to 
F/ as PE to PS, and therefore F/ 



. 
is equal to 



PE 



F/ is equal to Dd X 



; and DE 2 X 

DE 2 X PS 
PET~ ; 



and therefore the force of the la- 

DE 2 X PS 
mina EF/e is as Do? X PT?~ 

and the force of a particle exerted at the distance PF conjunctly ; that is, 
by the supposition, as DN X D(/ 7 or as the evanescent area DNwrf. 
Therefore the forces of all the lamina) exerted upon the corpuscle P are as 
all the areas DN//G?, that is, the whole force of the sphere will be as the 
whole area ANB. Q.E.D. 

COR. 1. Hence if the certripetal force tending to the several particles 

p)F 2 vx po 
remain always the same at all distances, and DN be made as ; 

Jr Jli 

the whole force with which the corpuscle is attracted by the sphere is as 
the area ANB. 

COR. 2. If the centripetal force of the particles be reciprocally as the 

DE 2 X PS 

distance of the corpuscle attracted by it, and DN be made as - ^^ , 

the force with which the corpuscle P is attracted by the whole sphere wil] 
be as the area ANB. 

Cor. 3. Jf the centripetal force of the particles be reciprocally as the 
cube of the distance of the corpuscle attracted by it, and DN be made as 

T)F 2 y PS 

--- . the force with which the corpuscle is attracted by the whole 

sphere will be as the area ANB. 

COR. 4. And universally if the centripetal force tending to the several 
particles of the sphere be supposed to be reciprocally as the quantity V ; 

DE 2 X PS 

and D5& be made as ^- ; the force with which a corpuscle is at- 

Jr Jtj X 

tracted by the whole sphere will be as the area ANB. 

PROPOSITION LXXXI. PROBLEM XLI. 

T/Le things remaining as above, it is required lo measure the area 

ANB. 

From the point P let there be drawn the right line PH touching the 
sphere in H ; and to the axis PAB, letting fall the perpendicular HI, 
bisect PI in L; and (by Prop. XII, Book II, El em.) PE 2 is equal tf 



SEC. XII.] 



OF NATURAL PHILOSOPHY. 



229 



PS 3 + SE 2 + 2PSD. But because 

the triangles SPH, SHI are alike, 

SE 2 or SH 2 is equal to the rectan 

gle PSI, Therefore PE 2 is equal 

to the rectangle contained under PS 

and PS -f SI + 2SD ; that is, under 

PS and 2LS + 2SD ; that is, under 

PS and 2LD. Moreover DE 2 is 

equal to SE 2 SD% or SE 2 

LS 2 + 2SLD LD 2 , that is, 2SLD LD 2 ALB. For LS- 

SE 2 or LS a SA a (by Prop. VI, Book II, Elem.) is equal to the rectan 

gle ALB. Therefore if instead of DE 2 we write 2SLD LD 2 ALB, 

the quantity - - ^-, which (by Cor. 4 of the foregoing Prop.) is as 




PE x 

the length of the ordinate DN, will 
2SLD x PS LD 2 X PS 



now 



resolve itself into three parts 



ALB xPS ... 

-TE3rr~ -pfixT" -pE^- v -; whereifinsteadofVwewnt 

the inverse ratio of the centripetal force, and instead of PE the mean pro 
portional between PS and 2LD, those three parts will become ordinates to 
so many curve lines, whose areas are discovered by the common methods. 
Q.E.D. 

EXAMPLE 1. If the centripetal force tending to the several particles of 
the sphere be reciprocally as the distance ; instead of V write PE the dis 
tance, then 2PS X LD for PE 2 ; and DN will become as SL LD 

ny |y Suppose DN equal to its double 2SL LD - r^ 5 an <* 2SL 

the given part of the ordinate drawn into the length AB will describe the 
rectangular area 2SL X AB ; and the indefinite part LD, drawn perpen 
dicularly into the same length with a continued motion, in such sort as in 
its motion one way or another it may either by increasing or decreasing re- 

LB 2 -LA 2 

main always equal to the length LD, will describe the area ^ , 

that is, the area SL X AB ; which taken from the former area 2SL X 

AB, leaves the area SL X AE. But the third part - ---, drawn after the 

i lit, 

same manner with a continued motion perpendicularly into the same length, 

will describe the area of an hyperbola, which subducted 

from the area SL X AB will leave ANB the area sought. 

Whence arises this construction of the Problem. At 

the points, L, A, B, erect the perpendiculars L/, Act, B6; 

making Aa equal to LB, and Bb equal to LA. Making 

L/ and LB asymptotes, describe through the points a, 6, 




230 



THE MATHEMATICAL PRINCIPLES 



[BOOK 1 



the hyperbolic crrve ab. And the chord ba being drawn, will inclose the 
area aba equal to the area sought ANB. 

EXAMPLE 2. If the centripetal force tending to the several particles of 
the sphere be reciprocally as the cube of the distance, or (which is the same 

PE 3 



thing; as that cube applied to any given plane ; write 
2PS X LD for PE 2 ; and DN will become as 



2AS 2 
SL X AS 2 



for V, and 

AS 2 



ALB X AS 2 
2PS X LD 2 
LSI 



PS X LD 2PS 

that is (because PS, AS, SI are continually proportional), as 
ALB X SI 



2LD : 
LSI 



If we draw then these three parts into th 



length AB, the first r-pr will generate the area of an hyperbola ; the sec- 

L-t \J 

, ALB X SI . ALB X SI 

ond iSI the area } AB X SI ; the third 2 Ll^ area - 2LA 



, that is, !AB X SI. From the first subduct the sum of the 
2LB 

second and third, and there will remain ANB, the area sought. Whence 
arises this construction of the problem. At the points L, A, S, B, erect 
the perpendiculars L/ Aa Ss, Bb, of which suppose Ss 
equal to SI ; and through the point s, to the asymptotes 
L/, LB, describe the hyperbola asb meeting the 
perpendiculars Aa, Bb, in a and b ; and the rectangle 
2ASI, subducted from the hyberbolic area AasbB, will 
B leave ANB the area sought. 




.. ,, . 

EXAMPLE 3. If the centripetal force tending to the several particles of 
the spheres decrease in a quadruplicate ratio of the distance from the par- 

pT^4 _ 

tides ; write ~|f- for V, then V 2PS + LD for PE, and DN will become 






___ 
V2SI X 




SI 2 X ALB 

2v2SI 



X 



These three parts drawn into the length AB, produce so many areas, viz. 



J-L 




2SI 2 X SL . 1 

x^ into 



T r 
LA 



~~~5ot in* V LB V LA; and 
BS1 2 X ALB . "1 1" 



VLA 3 v/LB 3 
And these after due reduction come 



fort h __ 



SEC. XII.] OF NATURAL PHILOSOPHY. 2 3\. 

2SI 3 4 SI 3 

~oj-p And these by subducting the last from the first, become -oT~r 

Therefore the entire force with ,7hich the corpuscle P is attracted towards 
the centre of the sphere is as-^, that is, reciprocally as PS 3 X PJ 

Q.E.I. 

By the same method one may determine the attraction of a corpuscle 
situate within the sphere, but more expeditiously by the following Theorem. 

PROPOSITION LXXXIL THEOREM XLI. 

In a sphere described about the centre S with the interval SA, if there be 
taken SI, SA, SP continually proportional ; ! sat/, that the attraction, 
of a corpuscle within the sphere in any place I is to its attraction without 
the sphere in the place P in a ratio compounded of the subduplicate 
ratio of IS, PS, the distances from the centre, and the subduplicate 
ratio of tJie centripetal forces tending to the centre in those places P 
and I. 

As if the centripetal forces of the 
particles of the sphere be reciprocally 
;is the distances of the corpuscle at 
tracted by them ; the force with which 
the corpuscle situate in I is attracted 
by the entire sphere will be to the 
force with which it is attracted in P 
in a ratio compounded of the subdu 
plicate ratio of the distance SI to the distance SP, and the subduplicate 
ratio of the centripetal force in the place I arising from any particle in the 
centre to the centripetal force in the place P arising from the same particle in 
the centre ; that is, in the subduplicate ratio of the distances SI, SP to each 
other reciprocally. These two subduplicate ratios compose the ratio of 
equality, and therefore the attractions in I and P produced by the whole 
sphere are equal. By the like calculation, if the forces of the particles of 
the sphere are reciprocally in a duplicate ratio of the distances, it will be 
found that the attraction in I is to the attraction in P as the distance SP 
to the semi -diameter SA of the sphere. If those forces are reciprocally in 
a triplicate ratio of the distances, the attractions in I and P will be to each 
other as SP 2 to SA 3 ; if in a quadruplicate ratio, as SP 3 to SA 3 . There 
fore since the attraction in P was found in this last case to be reciprocally 
as PS 3 X PI, the attraction in I will be reciprocally as S A 3 X PI, that is, 
because S A 3 is given reciprocally as PI. And the progression is the same 
in injinitnm. The demonstration of this Theorem is as follows : 

The things remaining as above constructed, and a corpuscle being in anj 




332 



THE MATHEMATICAL PRINCIPLES 



[BOOK I. 



place P. the ordinate DN was found to be as 



T)F 2 \" PS 

00 ^~\r- Therefore if 

I Cj X V 

IE be drawn, that ordinate for any other place of the corpuscle, as I, will 

DE 2 X IS 

become (mutatis mutandis] as ~T~p~rry~- Suppose the centripetalsorces 

flowing from any point of the sphere, as E, to be to each other at the dis 
tances IE and PE as PE 1 to IE 11 (where the number u denotes the index 

DE 2 X PS 

of the powers of PE and IE), and those ordinates will become as ^p - -57^7, 



2 \x IS 

and ~" --- TT7 , whose ratio to each other is as PS X IE X IE n to IS X 

IE X IE" 

PE X PE n . Because SI, SE, SP are in continued proportion, the tri 
angles SPE, SEI are alike ; and thence IE is to PE as IS to SE or SA. 
For the ratio of IE to PE write the ratio of IS to SA ; and the ratio of 
the ordinates becomes that of PS X IE" to SA X PE n . But the ratio of 
PS to SA is snbduplicate of that of the distances PS, SI ; and the ratio of 
IE" to PE 1 (because IE is to PE as IS to SA) is subduplicate of that of 
the forces at the distances PS, IS. Therefore the ordinates, and conse 
quently the areas whioifi the ordinates describe, and the attractions propor 
tional to them, are in a ratio compounded of those subduplicate ratios. 
Q.E.D. 

PROPOSITION LXXXIII. PROBLEM XLII. 

To find the force with which a corpuscle placed in the centre of a sphere 
is attracted towards any segment of that sphere whatsoever. 

Let P be a body in the centre of that sphere and 
RBSD a segment thereof contained under the plane 
RDS, and thesphrcrical superficies RBS. Let DB be cut 
in F by a sphaerical superficies EFG described from the 
centre P, and let the segment be divided into the parts 
_B BREFGS, FEDG. Let us suppose that segment to 
be not a purely mathematical but a physical superficies, 
having some, but a perfectly inconsiderable thickness. 
* Let that thickness be called O, and (by what Archi 
medes has demonstrated) that superficies will be as 
PF X DF X O. Let us suppose besides the attrac 
tive forces of the particles of the sphere to be reciprocally as that power of 
r.he distances, of which n is index ; and the force with which the superficies 

DE 2 X O 

EFG attracts the body P will be (by Prop. LXXIX) as -- that, 




2DF X O 
is, as ---? -,- 



DF 2 X O 
~"~ppn * 



pp n 

the perpendicular FN drawn into 



SEC. XJ11.I 



OF NATURAL PHILOSOPHY. 



233 



O be proportional to this quantity ; and the curvilinear area BDI, which 
the ordinate FN, drawn through the length DB with a continued motion 
will describe, will be as the whole force with which the whole segment 
RBSD attracts the body P. Q.E.I. 

PROPOSITION LXXXIV. PROBLEM XLIII. 

To find the force with which a corpuscle, placed without the centre of a 
sphere iti the axis of any segment, is attracted by that segment. 
Let the body P placed in. the axis ADB of 

the segment KBK be attracted by that seg 
ment. About the centre P, with the interval 

PE, let the spherical superficies EFK be de- 

scribed; and let it divide the segment into 

two parts EBKFE and EFKDE. Find the 

force of the first of those parts by Prop. 

LXXXI, and the force of the latter part by 

Prop. LXXXIII, and the sum of the forces will be the force of the whole 

segment EBKDE. Q.E.I. 

SCHOLIUM. 

The attractions of sphaerical bodies being now explained, it comes next 
in order to treat of the laws of attraction in other bodies consisting in like 
manner of attractive particles ; but to treat of them particularly is not neces 
sary to my design. It will be sufficient to subjoin some general proposi 
tions relating to the forces of such bodies, and the motions thence arising, 
because the knowledge of these will be of some little use in philosophical 
inquiries. 




SECTION XIII. 

Of the attractive forces of bodies which are not of a sphcerical figure. 

PROPOSITION LXXXV. THEOREM XLIL 

If a body be attracted by another, and its attraction be vastly stronger 
when it is contiguous to the attracting body than when they are sepa 
rated from one another by a very small interval ; the forces of the 
particles of the attracting body decrease, in the recess of the body at 
tracted, in more than a duplicate ratio of the distance of the particles. 
For if the forces decrease in a duplicate ratio of the distances from the 
particles, the attraction towards a sphaerical body being (by Prop. LXXIV) 
reciprocally as the square of the distance of the attracted body from the 
centre of the sphere, will not be sensibly increased by the contact, and it 



234 THE MATHEMATICAL PRINCIPLES [BOOK 1 

\vill be still less increased by it, if the attraction, in the recess of the body 
attracted, decreases in a still less proportion. The proposition, therefore, 
is evident concerning attractive spheres. And the case is the same of con 
cave sphaerical orbs attracting external bodies. And much more does it 
appear in orbs that attract bodies placed within them, because there the 
attractions diffused through the cavities of those orbs are (by Prop. LXX) 
destroyed by contrary attractions, and therefore have no effect even in the 
place of contact. Now if from these spheres and sphoerical orbs we take 
away any parts remote from the place of contact, and add new parts any 
where at pleas ore, we may change the figures of the attractive bodies at 
pleasure ; but the parts added or taken away, being remote from the place 
of contact, will cause no remarkable excess of the attraction arising from 
the contact of the two bodies. 1 herefore the proposition holds good in 
bodies of all figures. Q.E.I). 

PROPOSITION LXXXV1. THEOREM XLIII. 

If the forces of the particles of which an attractive body is composed de 
crease, in. the recess of the attractive body, in a triplicate or more than 
a triplicate ratio of the distance from the particles, the attraction will 
be vastly stronger in the point of contact than when the attracting and 
attracted bodies are separated from each other, though by never so 
small an interval. 

For that the attraction is infinitely increased when the attracted corpus 
cle comes to touch an attracting sphere of this kind, appears, by the solu 
tion of Problem XLI, exhibited in e second and third Examples. The 
same will also appear (by comparing those Examples and Theorem XLI 
together) of attractions of bodies made towards concavo-convex orbs, whether 
the attracted bodies be placed without the orbs, or in the cavities within 
them. And by aiding to or taking from those spheres and orbs any at 
tractive matter any where without the place of contact, so that the attrac 
tive bodies may receive any assigned figure, the Proposition will hold good 
of all bodies universally. Q.E.D. 

PROPOSITION LXXXVII. THEOREM XI. IV. 

If two bodies similar to each other, and consisting of matter equally at 
tractive attract separately two corpuscles proportional to those bodies, 
and in a like situation to them, the accelerative attractions of the cor 
puscles towards the entire bodies will be as the acccleratire at tractions 
of the corpuscles towards particles of the bodies proportional to the 
wholes, and alike situated in them. 

For if the bodies are divided into particles proportional to the wholes, 
and alike situated in them, it will be, as the attraction towards any parti 
cle of one of the bodies to the attraction towards the correspondent particle 



SEC. A III.] OF NATURAL PHILOSOPHY. 235 

in the other body, so are the attractions towards the several particles of the 
iirst body, to the attractions towards the several correspondent particles of 
the other body ; and, by composition, so is the attraction towards the first 
whole body to the attraction towards the second whole body. Q,.E.U. 

COR. 1 . Therefore if, as the distances of the corpuscles attracted increase, 
the attractive forces of the particles decrease in the ratio of any power 
of the distances, the accelerative attractions towards the whole bodies will 
be as the bodies directly, and those powers of the distances inversely. A* 
if the forces of the particles decrease in a duplicate ratio of the distances 
from the corpuscles attracted, and the bodies are as A 3 and B 3 , and there 
fore both the cubic sides of the bodies, and the distance of the attracted 
corpuscles from the bodies, are as A and B ; the accelerative attractions 

A 3 B 3 

towards the bodies will be as and , that is, as A and B the cubic 

sjides of those bodies. If the forces of the particles decrease in a triplicate 
ratio of the distances from the attracted corpuscles, the accelerative attrac- 

A 3 B 3 

tions towards the whole bodies will be as and 5--, that is, equal. If the 

A. tj 

forces decrease in a quadruplicate ratio, the attractions towards the bodies 

A 3 B 3 

will be as- an ^ 04 *^at is, reciprocally as the cubic sides A and B. 

And so in other cases. 

COR. 2. Hence, on the other hand, from the forces with which like bodies 
attract corpuscles similarly situated, may be collected the ratio of the de 
crease of the attractive forces of the particles as the attracted corpuscle 
recedes from them ; if so be that decrease is directly or inversely in any 
ratio of the distances. 

PROPOSITION LXXXVIII. THEOREM XLV. 

If the attractive forces of the equal particles of any body be as the dis 
tance of the places from the particles, the force of the whole body will 
tend to its centre of gravity ; and will be the same with the force of 
a globe, consisting of similar and equal matter, and having its centre 
in the centre of gravity. 
Let the particles A, B, of the body RSTV at 
tract any corpuscle Z with forces which, suppos-| 
ing the particles to be equal between themselves, 
are as the distances AZ, BZ ; but, if they are 
supposed unequal, are as those particles and 
their distances AZ, BZ, conjunctly, or (if I may 
go speak) as those particles drawn into their dis 
tances AZ, BZ respectively. And let those forces be expressed by the 




236 THE MATHEMATICAL PRINCIPLES [BOOK 1. 

contents u.ider A X AZ, and B X BZ. Join AB, and let it be cut in G, 
so that AG may be to BG as the particle B to the particle A : and G 
will be the common centre of gravity of the particles A and B. The force 
A X AZ will (by Cor. 2, of the Laws) be resolved into the forces A X GZ 
and A X AG ; and the force B X BZ into the forces B X GZ and B X 
BG. Now the forces A X AG and B X BG, because A is proportional to 
B, and BG to AG, are equal, and therefore having contrary directions de 
stroy one another. There remain then the forces A X GZ and B X GZ. 
These tend from Z towards the centre G, and compose the force A + B 
X GZ ; that is, the same force as if the attractive particles A and B were 
placed in their common centre of gravity G, composing there a little globe. 

By the same reasoning, if there be added a third particle G, and the 
force of it be compounded with the force A -f B X GZ tending to the cen 
tre G, the force thence arising will tend to the common centre of gravity 
of that globe in G and of the particle C ; that is, to the common centre oi 
gravity of the three particles A, B, C ; and will be the same as if that 
globe and the particle C were placed in that common centre composing a 
greater globe there ; and so we may go on in injinitum. Therefore 
the whole force of all the particles of any body whatever RSTV is the 
same as if that body, without removing its centre of gravity, were to put 
on the form of a globe. Q,.E.D. 

COR. Hence the motion of the attracted body Z will be the same as if 
the attracting body RSTV were sphaerical ; and therefore if that attract 
ing body be either at rest, or proceed uniformly in a right line, the body 
attracted will move in an ellipsis having its centre in the centre of gravity 
of the attracting body. 

PROPOSITION LXXXIX. THEOREM XLVI. 

If there be several bodies consisting of equal particles whose jorces are 
as the distances of the places from each, the force compounded of all 
the forces by which any corpuscle is attracted will tend to the common 
centre of gravity of the attracting bodies ; and will be the same as if 
those attracting bodies, preserving their common centre of gravity, 
should unite there, and be formed into a globe. 

This is demonstrated after the same manner as the foregoing Proposi 
tion. 

COR. Therefore the motion of the attracted body will be the same as if 
the attracting bodies, preserving their common centre of gravity, should 
unite there, and be formed into a globe. And, therefore, if the common 
centre of gravity of the attracting bodies be either at rest, or proceed uni 
formly in a right line, the attracted body will move in an ellipsis having 
Us centre in the common centre of gravity of the attracting bodies. 



SEC. XlII.j OF NATURAL PHILOSOPHY. 237 

PROPOSITION XC. PROBLEM XLIV. 

If to the several points of any circle there tend equal centripeta forces, 
increasing or decreasing in any ratio of the distances ; it is required 
to Jin d the force with which a corpuscle is attracted, that is, situate 
any where in a right line which stands at right angles to the plant 
of the circle at its centre. 
Suppose a circle to be described about the cen 
tre A with any interval AD in a plane to which ; 
the right line AP is perpendicular ; and let it be 
required to find the force with which a corpuscle 
P is attracted towards the same. From any point 
E of the circle, to the attracted corpuscle P, let 
there be drawn the right line PE. In the right 
line PA take PF equal to PE, and make a per- 
pendicular FK, erected at F, to be as the force 
with which the point E attracts the corpuscle P. 
And let the curve line IKL be the locus of the point K. Let that cu/, fe 
meet the plane of the circle in L. In PA take PH equal to PD, and p/^ct 
the perpendicular HI meeting that curve in I ; and the attraction of the 
corpuscle P towards the circle will be as the area AHIL drawn into the 
altitude AP Q.E.I. 

For let there be taken in AE a very small line Ee. Join Pe, and in PE, 
PA take PC, Pf equal to Pe. And because the force, with which any 
point E of the annulus described about the centre A with the interval AS 
in the aforesaid plane attracts to itself the body P, is supposed to be as 
FK ; and, therefore, the force with which that point attracts the body P 

AP X FK 

towards A is as - ^p ; and the force with which the whole annulus 

AP X FK 

attracts tne body P towards A is as the annulus and p^ conjunct- 

ly ; and that annulus also is as the rectangle under the radius AE aad the 
breadth Ee, and this rectangle (because PE and AE, Ee and CE are pro 
portional) is equal to the rectangle PE X CE or PE X F/; the force 
*-ith which that annulus attracts the body P towards A will be as PE X 

AP X FK 
Ff and pp~~~ conjunctly ; that is, as the content under F/ X FK X 

AP, or as the area FKkf drawn into AP. And therefore the sum of the 
forces with which all the annuli, in the circle described about the centre A 
with the interval AD, attract the body P towards A, is as the whole area 
AHIKL drawn into AP. Q.E.D. 
COR. 1. Hence if the forces of the points decrease in the duplicate ratio 



238 



THE MATHEMATICAL PRINCIPLES 



[BOOK I 



of the distances, that is, if FK be as rfFK, and therefore the area AHIKL 



as p-7 p- ; the attraction of the corpuscle P towards the circle will 



PA AH 

be as 1 ; that is, as 



COR. 2. And universally if the forces of the points at the distances D b( 
reciprocally as any power D n of the distances; that is, if FK be as . 



and therefore the area AHIKL as 



1 



1 



" l PH" 

1 PA 



, ; the attraction 



of the corpuscle P towards the circle will be as 

PA" 2 PH" l 

COR. 3. And if the diameter of the circle be increased in itifinitum, and 
the number n be greater than unity ; the attraction of the corpuscle P to 
wards the whole infinite plane will be reciprocally as PA" 2 , because the 

PA 

other term vanishes. 

PROPOSITION XCI. PROBLEM XLV. 

To find the attraction of a corpuscle situate in the axis of a round solid, 
to whose several points there tend equal centripetal forces decreasing 
in any ratio of the distances whatsoever. 

Let the corpuscle P, situate in the axis AB 
of the solid DECG, be attracted towards that 
solid. Let the solid be cut by any circle as 
RFS, perpendicular to the axis ; and in its 
semi-diameter FS, in any plane PALKB pass 
ing through the axis, let there be taken (by 
Prop. XC) the length FK proportional to the 
force with which the corpuscle P is attracted 
towards that circle. Let the locus of the point 
K be the curve line LKI, meeting the planes of the outermost circles AL 
and BI in L and I ; and the attraction of the corpuscle P towards the 
solid will be as the area LABI. Q..E.I. 

COR. 1. Hence if the solid be a cylinder described by the parallelogram 
A DEB revolved about the axis AB, and the centripetal forces tending to 
the several points be reciprocally as the squares of the distances from the 
points ; the attraction of the corpuscle P towards this cylinder will be as 
AB PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will be 

PF 

as 1 --. The part 1 of this quantity, drawn into the length AB, de- 




SEC. XIII. 



OF NATURAL PHILOSOPHY 



239 



scribes the area 1 X AB ; and the other part 

PF 

, drawn into the length PB describes the 



ix 



area 1 into PE AD (as may be easily 
shewn from the quadrature of the curve 
LKI); and, in like manner, the same part 
drawn into the length PA describes the area 

L into PD AD. and drawn into AB, the 



"At 



G 



Iv 



S 



13 



M 

7J" 




1 



difference of PB and PA, describes 1 into PE PD, the difference of the 
areas. From the first content 1 X AB take away the last content 1 into 
PE PD, and there will remain the area LABI equal to 1 into 
AB PE -h PD. Therefore the force, being proportional to this area, 
is as AB PE + PD. 

COR. 2. Hence also is known the force 
by which a spheroid AGBC attracts any 
body P situate externally in its axis AB. 
Let NKRM be a conic section whose or- 
dinate KR perpendicular to PE may be \ 
always equal to the length of the line PD, 
continually drawn to tlie point D in 
which that ordinate cuts the spheroid. 
From the vertices A, B, of the spheriod, 
let there be erected to its axis AB the perpendiculars AK, BM, respectively 
equal to AP. BP, and therefore meeting the conic section in K and M; and 
join KM cutting off from it the segment KMRK. Let S be the centre of the 
spheroid, and SC its greatest semi-diameter : and the force with which the 
spheroid attracts the body P will be to the force with which a sphere describ- 

, ....,,. ASxCS 2 -PSxKMRK 

ed with the diameter AhJ attracts the same body as prrr ^ r-= 

1 o -f- Go 2 Ao 

AS 3 
is to fkT ^,. And by a calculation founded on the same principles may be 



found the forces of the segments of the spheroid. 

COR. 3. If the corpuscle be placed within the spheroid and in its axis, 
the attraction will be as its distance from the centre. This may be easily 
collected from the following reasoning, whether 
the particle be in the axis or in any other given 
diameter. Let AGOF be an attracting sphe 
roid, S its centre, and P the body attracted. 
Through the body P let there be drawn the 
semi-diameter SPA, and two right lines DE, 
FC meeting the spheroid in 1) and E, F and 
G ; and let PCM, HLN be the superficies of 




240 THE MATHEMATICAL PRINCIPLE* ffioOK 1. 

two interior spheroids similar and concentrical to the exterior, the first of 
which passes through the body P. and cuts the right lines DE, FG in B 
and C ; arid the latter cuts the same right lines in H and I, K and L. 
I ,et the spheroids have all one common axis, and the parts of the right 
lines intercepted on both sides DP and BE, FP and CG, DH and IE, FK 
and LG, will be mutually equal; because the right lines DE. PB, and HI. 
are bisected in the same point, as are also the right lines FG, PC, and KL. 
Conceive now DPF. EPG to represent opposite cones described with the 
infmitely small vertical angles DPF, EPG, and the lines DH, El to be 
infinitely small also. Then the particles of the cones DHKF, GLIE, cut 
off by the spheroidical superficies, by reason of the equality of the lines DH 
and ET ; will be to one another as the squares of the distances from the body 
P, and will therefore attract that corpuscle equally. And by a like rea 
soning if the spaces DPF, EGCB be divided into particles by the superfi 
cies of innumerable similar spheroids concentric to the former and having 

J. O 

one common axis, all these particles will equally attract on both sides the 
body P towards contrary parts. Therefore the forces of the cone DPF. 
and of the conic segment EGCB, are equal, and by their contrariety de 
stroy each other. And the case is the same of the forces of all the matter 
that lies without the interior spheroid PCBM. Therefore the body P is 
attracted by the interior spheroid PCBM alone, and therefore (by Cor. 3, 
Prop. 1 , XXII) its attraction is to the force with which the body A is at 
tracted by the whole spheroid AGOD as the distance PS to the distance 
AS. Q.E.D. 

PROPOSITION XCII. PROBLEM XLVI. 

An attracting body being given, it is required to find the ratio of the de 
crease of the centripetal forces tending to its several points. 
The body given must be formed into a sphere, a cylinder, or some regu 
lar figure, whose law of attraction answering to any ratio of decrease may 
be found by Prop. LXXX, LXXXI, and XCI. Then, by experiments, 
the force of the attractions must be found at several distances, and the law 
of attraction towards the whole, made known by that means, will give 
the ratio of the decrease of the forces of the several parts ; which was to 
be found. 

PROPOSITION XCIII. THEOREM XLVII. 

If a solid be plane on one side, and infinitely extended on all otljer sides, 
and consist of equal particles equally attractive, iv hose forces decrease, 
in the recess from the solid, in the ratio of any power greater than the 
square of the distances ; and a corpuscle placed towards eit far part of 
the plane is attracted by the force of the whole solid ; I say that tfie 
attractive force of the whole solid, in the recess from its platw superfi- 



XIILj 



OF NATURAL PHILOSOPHY". 



241 



n 



H 



m 



G 



ties, will decrease in the ratio of a power whose side is the distance oj 

the corpuscle from the plane, and its index less by 3 than the index oj 

the power of the distances. 

CASE 1. Let LG/be the plane by which 
the solid is terminated. Let the solid 
lie on that hand of the plane that is to 
wards I, and let it be resolved into in- _. 
numerable planes mHM, //IN, oKO, 
(fee., parallel to GL. And first let the 
attracted body C be placed without the 
solid. Let there be drawn CGHI per 
pendicular to those innumerable planes, 
and let the attractive forces of the points of the solid decrease in the ratio 
of a power of the distances whose index is the number n not less than 3. 
Therefore (by Cor. 3, Prop. XC) the force with which any plane mHM 
attracts the point C is reciprocally as CH n 2 . In the plane mHM take the 
length HM reciprocally proportional to CH 1 2 , and that force will be as 
HM. In like manner in the several planes /GL, //,TN, oKO, (fee., take the 
lengths GL, IN, KO, (fee., reciprocally proportional to CG n 2 , CI 1 2 , 
CK n 2 , (fee., and the forces of those planes will bs as the lengths so taken, 
and therefore the sum of the forces as the sum of the lengths, that is, the 
force of the Avhole solid as the area GLOK produced infinitely towards 
OK. But that area (by the known methods of quadratures) is reciprocally 
as CG n 3 , and therefore the force of the whole solid is reciprocally as 
CG"- 3 . Q.E.D. 

CASE 2. Let ttecorpuscleC be now placed on that 
hand of the plane /GL that is within the solid, 
and take the distance CK equal to the distance 
CG. And the part of the solid LG/oKO termi 
nated by the parallel planes /GL, oKO, will at 
tract the corpuscle C, situate in the middle, neither 
one way nor another, the contrary actions of the 
opposite points destroying one another by reason of 
their equality. Therefore the corpuscle C is attracted by the force only 
of the solid situate beyond the plane OK. But this force (by Case 1) is 
reciprocally as CK n 3 , that is, (because CG, CK are equal) reciprocally as 
CG" 3 . Q,.E.D. 

COR. 1. Hence if the solid LGIN be terminated on each sitfe by two in 
finite parallel places LG, IN, its attractive force is known, subducting 
from the attractive force of the whole infinite solid LGKO the attractive 
force of the more distant part NIKO infinitely produced towards KO. 

COR. 2. If the more distant part of this solid be rejected, because its at 
traction compared with the attraction of the nearer part is inconsiderable, 

16 





N 








K 1 



C 





242 THE MATHEMATICAL PRINCIPLES [BOOK 1 

the attraction of that nearer part will, as the distance increases, decrease 
nearly in the ratio of the power CG n 3 . 

Con. 3. And hence if any finite body, plane on one side, attract a cor 
puscle situate over against the middle of that plane, and the distance between 
the corpuscle and the plane compared with the dimensions of the attracting 
body be extremely small ; and the attracting body consist of homogeneous 
particles, whose attractive forces decrease in the ratio of any power of the 
distances greater than the quadruplicate ; the attractive force of the whole 
body will decrease very nearly in the ratio of a power whose side is that 
very small distance, and the index less by 3 than the index of the former 
power. This assertion does not hold good, however, of a body consisting 
of particles whose attractive forces decrease in the ratio of the triplicate 
power of the distances ; because, in that case, the attraction of the remoter 
part of the infinite body in the second Corollary is always infinitely greater 
than the attraction of the nearer part. 



SCHOLIUM. 

If a body is attracted perpendicularly towards a given plane, and from 
the law of attraction given, the motion of the body be required ; the Pro 
blem will be solved by seeking (by Prop. XXXIX) the motion of the body 
descending in a right line towards that plane, and (by Cor. 2, of the Laws) 
compounding that motion with an uniform motion performed in the direc 
tion of lines parallel to that plane. And, on the contrary, if there be re 
quired the law of the attraction tending towards the plane in perpendicu 
lar directions, by which the body may be caused to move in any given 
curve line, the Problem will be solved by working after the manner of the 
third Problem. 

But the operations may be contracted by resolving the ordinates into 
converging series. As if to a base A the length B be ordinately ap 
plied in any given angle, and that length be as any power of the base 

A^ ; and there be sought the force with which a body, either attracted to 
wards the base or driven from it in the direction of that ordinate, may be 
caused to move in the curre line which that ordinate always describes with 
its superior extremity ; I suppose the base to be increased by a very small 

,m m 

part O, and I resolve the ordinate A -f Ol^ into an infinite series A- -f 

!!L OA ^ + ^-^--- OOA ;- &c., and I suppose the force proper- 
11 1111 

tional to the term of this series in which O is of two dimensions, that is, 
to the term - - OOA ^YT, Therefore the force sought is aa 



SEC. XIV.J OF NATURAL PHILOSOPHY. 2M 

mm mn m 2n .... mm mn m 2n 

A 7, , or, which is the same thinor, as L> m . 

nn nn 

As if the ordinate describe a parabola, m being 2, and n = 1, the force 
will be as the given quantity 2B, and therefore is given. Therefore with 
a given force the body will move in a parabola, as Galileo has demon 
strated. If the ordinate describe an hyperbola, m being = 1, and n 
1, the force will be as 2 A 3 or 2B 3 ; and therefore a force which is as the 
cube of the ordinate will cause the body to move in an hyperbola. But 
leaving this kind of propositions, I shall go on to some others relating to 
motion which I have fiot yet touched upon. 



SECTION XIV. 

Of the motion of very small bodies when agitated by centripetal forces 
tending to the several parts of any very great body. 

PROPOSITION XCIV. THEOREM XLVIII. 

If two similar mediums be separated from each other by a space termi 
nated on both sides by parallel planes, and a body in its passage 
through that space be attracted or impelled perpendicularly towards 
either of those mediums, and not agitated or hindered by any other 
force ; and the attraction be every where the same at equal distances 
from either plane, taken towards the same hand of the plane ; I say, 
that the sine of incidence upon either plane will be to the sine of emcr 
gence from the other plane in a given ratio. 
CASE 1. Let Aa and B6 be two parallel planes, 
and let the body light upon the first plane Aa in 
the direction of the line GH, and in its whole 
passage through the intermediate space let it be 
attracted or impelled towards the medium of in 
cidence, and by that action let it be made to de 
scribe a curve line HI, and let it emerge in the di 
rection of the line IK. Let there be erected IM 
perpendicular to Eb the plane of emergence, and 
meeting the line of incidence GH prolonged in M, and the plane of inci 
dence Aa in R ; and let the line of emergence KI be produced and meet 
HM in L. About the centre L, with the interval LI, let a circle be de 
scribed cutting both HM in P and Q, and MI produced in N ; and, first, 
if the attraction or impulse be supposed uniform, the curve HI (by what 
Galileo has demonstrated) be a parabola, whose property is that of a roc- 




THE MATHEMATICAL PRINCIPLES [BoOK 1 

tangle under its given latiis rectum and the line IM is equal to the squarrf 
cf HM ; and moreover the line HM will be bisected in L. Whence if to 
MI there be let fall the perpendicular LO, MO, OR will be equal; and 
adding the equal lines ON, OI, the wholes MN, IR will be equal also. 
Therefore since IR is given, MN is also given, and the rectangle NMI is 
to the rectangle under the latus rectum and IM, that is, to HM a in a given 
ratio. But the rectangle NMI is equal to the rectangle PMQ,, that is, to 
the difference of the squares ML 2 , and PL 2 or LI 2 ; and HM 2 hath a given 
ratio to its fourth part ML 2 ; therefore the ratio of ML 2 LI 2 to ML 2 is given, 
and by conversion the ratio of LI 2 to ML , and its subduplicate, theratrio 
of LI to ML. But in every triangle, as LMI, the sines jf the angles are 
proportional to the opposite sides. Therefore the ratio of the sine of the 
angle of incidence LMR to the sine of the angle of emergence LIR is 
given. QJE.lr). 

CASE 2. Let now the body pass successively through several spaces ter 
minated with parallel planes Aa/>B, B6cC, &c., and let it be acted on by a 
\ . force which is uniform in each of them separ- 

\ a ately, but different in the different spaces ; and 

B \ fr by what was just demonstrated, the sine of the 

c ^^ c angle of incidence on the first plane Aa is to 

the sine of emergence from the second plane Bb 

in a given ratio ; and this sine of incidence upon the second plane Bb will 
be to the sine of emergence from the third plane Cc in a given ratio ; and 
this sine to the sine of emergence from the fourth plane Dd in a given ra 
tio ; and so on in infinitum ; and, by equality, the sine of incidence on 
the first plane to the sine of emergence from the last plane in a given ratio. 
I ,et now the intervals of the planes be diminished, and their number be in 
finitely increased, so that the action of attraction or impulse, exerted accord 
ing to any assigned law, may become continual, and the ratio of the sine of 
incidence on the first plane to the sine of emergence from the last plane 
being all along given, will be given then also. QJE.D. 

PROPOSITION XCV. THEOREM XLIX. 

The same thing s being supposed, I say, that the velocity of the body be 
fore its incidence is to its velocity after emergence as the sine of emer 
gence to the sine of incid nee. 

Make AH and Id equal, and erect the perpen- 
diculars AG, dK meeting the lines of incidence 
and emergence GH, IK, in G and K. In GH 
-- take TH equal to IK, and to the plane Aa let 
^ fall a perpendicular TV. And (by Cor. 2 of the 
|x^ I Laws of Motion) let the motion of the body be 
j v - resolved into two, one perpendicular to the planes 




SEC. X1V.J OF NATURAL PHILOSOPHY. 245 

Aa, Bb, Cc, &c, and another parallel to them. The force of attraction or 
impulse, acting in directions perpendicular to those planes, does not at all 
alter the motion in parallel directions ; and therefore the body proceeding 
with this motion will in equal times go through those equal parallel inter 
vals that lie between the line AG and the point H, and between the point 
I and the line dK ; that is, they will describe the lines GH, IK in equal 
times. Therefore the velocity before incidence is to the velocity after 
emergence as GH to IK or TH, that is, as AH or Id to vH, that is (sup 
posing TH or IK radius), as the sine of emergence to the sine of inci 
dence. Q.E.D. 

PROPOSITION XCVL THEOREM L. 

The same things being supposed, and that the motion before incidence is 
swifter than afterwards ; 1 sat/, lhat if the line of incidence be in 
clined continually, the body will be at last reflected, and the angle of 
reflexion will be equal to the angle of incidence. 

For conceive the body passing between the parallel planes Aa, Bb, Cc, 
&c., to describe parabolic arcs as above; 
and let those arcs be HP, PQ, QR, &c. 
And let the obliquity of the line of inci- g 
dence GH to the first plane Aa be such rc~ 

that the sine of incidence may be to the radius of the circle whose sine it is, 
in the same ratio which the same sine of incidence hath to the sine of emer 
gence from the plane Dd into the space DefeE ; and because the sine of 
emergence is now become equal to radius, the angle of emergence will be a 
right one, and therefore the line of emergence will coincide with the plane 
Dd. Let the body come to this plane in the point R ; and because the 
line of emergence coincides with that plane, it is manifest that the body can 
proceed no farther towards the plane Ee. But neither can it proceed in the 
line of emergence Rd; because it is perpetually attracted or impelled towards 
the medium of incidence. It will return, therefore, between the planes Cc, 
Dd, describing an arc of a parabola Q,R</, whose principal vertex (by what 
Galileo has demonstrated) is in R, cutting the plane Or in the same angle 
at q, that it did before at Q, ; then going on in the parabolic arcs qp, ph, 
&c., similar and equal to the former arcs QP, PH, &c., it will cut the rest 
of the planes in the same angles at p, h, (fee., as it did before in P, H, (fee., 
and will emerge at last with the same obliquity at h with which it first 
impinged on that plane at H. Conceive now the intervals of the planes 
Aa, Bb, Cc, Dd, Ee, (fee., to be infinitely diminished, and the number in 
finitely increased, so that the action of attraction or impulse, exerted ac 
cording to any assigned law, may become continual; and, the angle of 
emergence remaining all alor g equal to the angle of incidence, will be 
equal to the same also at last. Q.E.D. 



246 THE MATHEMATICAL PRINCIPLES IBoOK 1 

SCHOLIUM. 

These attractions bear a great resemblance to the reflexions and refrac 
tions of light made in a given ratio of the secants, as was discovered h} 
Siiellius ; and consequently in a given ratio of the sines, as was exhibited 
by Hes Cortes. For it is now certain from the phenomena of Jupiter s 
^satellites, confirmed by the observations of different astronomers, that light 
is propagated in succession, and requires about seven or eight minutes to 
travel from the sun to the earth. Moreover, the rays of light that are in 
our air (as lately was discovered by Grimaldus, by the admission of light 
into a dark room through a small hole, which 1 have also tried) in their 
passage near the angles of bodies, whether transparent or opaque (such aa 
the circular and rectangular edges of gold, silver and brass coins, or of 
knives, or broken pieces of stone or glass), are bent or inflected round those 
bodies as if they were attracted to them ; and those rays which in their 
passage come nearest to the bodies are the most inflected, as if they were 
most attracted : which tiling I myself have also carefully observed. And 
those which pass at greater distances are less inflected ; and those at still 
greater distances are a little inflected the contrary way, and form three 
fringes of colours. In the figure 5 represents the edge of a knife, or any 



tl \ 


/ 


K", \ 


/ 


^::^ 

x -^ . v 


^--:<*, 



-f:: : r ; ^c :.-/ 

N >V J 



V U 

W~"~a~" "~a C: O la 



kind of wedge AsB : and gowog,fmnif,emtme, dlsld, are rays inflected to 
wards the knife in the arcs owo, nvn, mtm, Isl ; which inflection is greater 
or less according; to their distance from the knife. Now since this inflec 
tion of the rays is performed in the air without the knife, it follows that the 
rays which fall upon the knife are first inflected in the air before they touch 
the knife. And the case is the same of the rays falling upon glass. The 
refraction, therefore, is made not in the point of incidence, but gradually, by 
a continual inflection of the rays ; which is done partly in the air before they 
touch the glass, partly (if [ mistake not) within the glass, after they have 
entered it ; as is represented in the rays ckzc, bujb^ ahxa, falling upon r, 
q, p, and inflected between k and z, i and y, h and x. Therefore because 
of the analogy there is between the propagation of the rays f light and the 
motion of bodies, I thought it not amiss to add the followi g Propositions 
far optical uses ; not at all. considering the nature of the rays of .light, or 
inquiring whether they are bodies or not ; but only determining the tra 
jectories of bodies which are extremely like the trajectories of the rays. 




SEC. XIV.] OF NATURAL PHILOSOPHY. 247 

PROPOSITION XCVII. PROBT.-EM XL VII. 
Supposing t/w sine of incidence upon any superficies to be in a given ra 

tio to the sine of emergence ; and that tha inflection of t/ts paths of 

those bodies near that superficies is performed in a very short space, 

which may be considered as a point ; it is required to determine suck 

a superficies as may cause all the corpuscles issuing from any one 

given place to converge to another given place. 

Let A be the place from whence the cor- E 

puscles diverge ; B the place to which they 

should converge ; CDE the curve line which 

by its revolution round the axis AB describes . /C 

the superficies sought ; D, E, any two points of that curve ; and EF, EG, 

perpendiculars let fall on the paths of the bodies AD, DB. Let the point 

D approach to and coalesce with the point E ; and the ultimate ratio of 

the line DF by which AD is increased, to the line DG by which DB is 

diminished, will be the same as that of the sine of incidence to the sine of 

emergence Therefore the ratio of the increment of the line AD to the 

decrement of the line DB is given: and therefore if in the axis AB there 

be taken any where the point C through which the curve CDE must 

pass, and CM the increment of AC be taken in that given ratio to CN 

the decrement of BC, and from the centres A, B, with the intervals AM, 

BN, there be described two circles cutting each other in D ; that point D 

will touch the curve sought CDE, and, by touching it any where at pleasure, 

will determine that curve. Q.E.I. 

COR. 1. By causing the point A or B to go off sometimes in infinitum, 
and sometimes to move towards other parts of the point C, will bo obtain 
ed all those figures which Cartesins has exhibited in his Optics and Geom 
etry relating to refractions. The invention of which Cartcsius having 
thought fit to conceal, is here laid open in this Proposition. 

COR. 2. If a body lighting on any superfi 
cies CD in the direction of a ri^ht line AD, Qj- 

O \ 

drawn according to any law, should emerge 
in the direction of another right line DK ; 
and from the point C there be drawn curve 




lines CP, CQ, always perpendicular to AD, DK ; the increments of the 
lines PD, QD, and therefore the lines themselves PD, Q.D, generated by 
those increments, will be as the sines of incidence and emergence to each 
other, and e contra. 

PROPOSITION XCVIII. PROBLEM XLVIII. 

The same things supposed ; if round the axis AB any attractive super 
ficies be described as CD, regular or irregular, through which the bo 
dies issuing from the given place A must pass ; it is required to find 




THE MATHEMATICAL PRINCIPLES. [BOOK J 

a second attractive superficies EF, which may make those bodies con 
verge to a given place B. 

Let a line joining AB cut 
the first superficies in C and 
the second in E, the point D 
being taken any how at plea 
sure. And supposing the 
f sine of incidence on the first 
superficies to the sine of 
emergence from the same, and the sine of emergence from the second super 
ficies to the sine of incidence on the same, to be as any given quantity M 
to another given quantity N; then produce AB to G, so that BG may be 
to CE as M N to N ; and AD to H, so that AH may be equal to AG ; 
arid DF to K, so that DK may be to DH as N to M. Join KB, and about 
the centre D with the interval DH describe a circle meeting KB produced 
in L, and draw BF parallel to DL; and the point F will touch the line 
EF, which, being turned round the axis AB, will describe the superficies 
sought. Q.E.F. 

For conceive the lines CP, CQ to be every where perpendicular to AD, 
DF, and the lines ER, ES to FB, FD respectively, and therefore QS to 
be always equal to CK; and (by Cor. 2, Prop. XCVII) PD will be to QD 
as M to N, and therefore as DL to DK, or FB to FK ; and by division as 
DL FB or PH PD FB to FD or FQ QD ; arid by composition 
as PH FB to FQ, that is (because PH and CG, QS and CE, are equal), 
as CE + BG FR to CE FS. But (because BG is to CE as M 
N to N) it. comes to pass also that CE + BG is to CE as M to N; and 
therefore, by division, FR is to FS as M to N ; and therefore (by Cor. 2, 
Prop XCVI1) the superficies EF compels a body, falling upon it in the 
direction DF, to go on in the line FR to the place B. Q.E.D. 

SCHOLIUM. 

,In the same manner one may go on to three or more superficies. But 
of all figures the sphserical is the most proper for optical uses. If the ob 
ject glasses of telescopes were made of two glasses of a sphaerical figure, 
containing water between them, it is not unlikely that the errors of the 
refractions made in the extreme parts of the superficies of the glasses may 
be accurately enough corrected by the refractions of the water. Such ob 
ject glasses are to be preferred before elliptic and hyperbolic glasses, not only 
because they may be formed with more ease and accuracy, but because the 
pencils of rays situate without the axis of the glass would be more accu 
rately refracted by them. But the different refrangibility of different raya 
is the real obstacle that hinders optics from being made perfect by sphaeri 
cal or any other figures. Unless the errors thence arising can be corrected, 
all the labour spent in correcting the others is quite thrown away. 



BOOK II 



BOOK II. 



OF THE MOTION OF BODIES. 

SECTION I. 
Of the motion of bodies that are resisted in the ratio of the velocity. 

PROPOSITION I. THEOREM I. 

Tf a body is resisted in the ratio of its velocity, the motion lost by re 
sistance is as the space gone over in its motion. 

For since the motion lost in each equal particle of time is as the velocity, 
that is, as the particle of space gone over, then, by composition, the motion 
lost in the whole time will he as the whole space gone over. Q.E.D. 

COR. Therefore if the body, destitute of all gravity, move by its innate 
force only in free spaces, and there be given both its whole motion at the 
beginning, and also the motion remaining after some part of the way is 
gone over, there will be given also the whole space which the body can de 
scribe in an infinite time. For that space will be to the space now de 
scribed as the whole motion at the beginning is to the part lost of that 
motion. 

LEMMA I. 

Quantities proportional to their differences are continually proportional. 
Let A be to A B as B to B C and C to C D, (fee., and, by con 
version, A will be to B as B to C and C to D, &c. Q.E.D. 

PROPOSITION II. THEOREM II. 

If a body is resisted in the ratio of its velocity, and moves, by its vis in- 
sita only, through a similar medium, and the times be taken equal, 
the velocities in the beginning of each of the times are in a geometri 
cal progression, and the spaces described in each of the times are as 
the velocities. 

CASE 1. Let the time be divided into equal particles ; and if at the very 
beginning of each particle we suppose the resistance to act with one single 
impulse which is as the velocity, the decrement of the velocity in each of 



THE MATHEMATICAL PRINCIPLES [BOOK II. 

the particles of time will be as the same velocity. Therefore the veloci 
ties are proportional to their differences, and therefore (by Lem. 1, Book 
II) continually proportional. Therefore if out of an equal number of par 
ticles there be compounded any equal portions of time, the velocities at the 
beginning of those times will be as terms in a continued progression, which 
are taken by intervals, omitting every where an equal number of interme 
diate terms. But the ratios of these terms are compounded of the equa j 
ratios of the intermediate terms equally repeated, and therefore are equal 
Therefore the velocities, being proportional to those terms, are in geomet 
rical progression. Let those equal particles of time be diminished, and 
their number increased in infinitum, so that the impulse of resistance may 
become continual ; and the velocities at the beginnings of equal times, al 
ways continually proportional, will be also in this case continually pro 
portional. Q.E.D. 

CASE 2. And, by division, the differences of the velocities, that is, the 
parts of the velocities lost in each of the times, are as the wholes ; but the 
spaces described in each of the times are as the lost parts of the velocities 
(by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D. 
TT COROL. Hence if to the rectangular asymptotes AC, CH, 

the hyperbola BG is described, and AB, DG be drawn per 
pendicular to the asymptote AC, and both the velocity of 
. the body, and the resistance of the medium, at the very be 
ginning of the motion, be expressed by any given line AC, 
and, after some time is elapsed, by the indefinite line DC ; the time may 
be expressed by the area ABGD, and the space described in that time by 
the line AD. For if that area, by the motion of the point D, be uniform 
ly increased in the same manner as the time, the right line DC will de 
crease in a geometrical ratio in the same manner as the velocity ; and the 
parts of the right line AC, described in equal times, will decrease in the 
same ratio. 

PROPOSITION III. PROBLEM I. 

To define the motion of a body which, in a similar medium, ascends or 
descends in a right line, and is resisted in the ratio of its velocity, and 
acted upon by an uniform force of gravity. 

The body ascending, let the gravity be expound 
ed by any given rectangle BACH ; and the resist 
ance of the medium, at the beginning of the ascent, 
by the rectangle BADE, taken on the contrary side 
Jfl e B^l | L- of the right line AB. Through the point B, with 
the rectangular asymptotes AC, CH, describe an 
hyperbola, cutting the perpendiculars DE, de, ID 






SEC. I.j OF NATURAL PHILOSOPHY. 253 

G, g ; and the body ascending will in the time DGgd describe the space 
EG-e; in the time DGBA, the space of the whole ascent EGB ; in the 
time ABK1, the space of descent BFK ; and in the time IKki the space of 
descent KFfk; and the velocities of the bodies (proportional to the re 
sistance of the medium) in these periods of time will be ABED, ABed, O, 
ABFI, AB/z respectively ; and the greatest velocity which the body can 
acquire by descending will be BACH. 

For let the rectangle BACH be resolved into in 
numerable rectangles AA , K/, Lm, M//, *fea, which 
shall be as the increments of the velocities produced 
in so many equal times; then will 0, AAr, AL Am, An, 
ifec., be as the whole velocities, and therefore (by suppo 
sition) as the resistances of the medium in the be- 

ginning of each of the equal times. Make AC to AJLLB 

AK, or ABHC to AB/vK, as the force of gravity to the resistance in the 
beginning of the second time ; then from the force of gravity subduct the 
resistances, and ABHC, K/vHC, L/HC, MwHC, (fee., will be as the abso 
lute forces with which the body is acted upon in the beginning of each of 
the times, and therefore (by Law I) as the increments of the velocities, that 
is, as the rectangles AA-, K/, Lm, M//, (fee., and therefore (by Lem. 1, Book 
II) in a geometrical progression. Therefore, if the right lines K, LI 
M/TO, N//, &c., are produced so as to meet the hyperbola in q, r, s, t, (fee.. 
the areas AB^K, K</rL, LrsM, MsJN, (fee., will be equal, and there 
fore analogous to the equal times and equal gravitating forces. But the 
area AB^K (by Corol. 3, Lem. VII and VIII, Book I) is to the area Bkq 
as K^ to \kq, or AC to |AK, that is, as the force of gravity to the resist 
ance in the middle of the first time. And by the like reasoning, the areas 
<?KLr, rLMs, sMN/, (fee., are to the areas qklr, rims, smnt, (fee., as the 
gravitating forces to the resistances in the middle of the second, third, fourth 
time, and so on. Therefore since the equal areas BAKq, </KLr, rLMs, 
sMN/, (fee., are analogous to the gravitating forces, the areas Bkq, qklr, 
rims, smut, (fee., will be analogous to the resistances in the middle of 
each of the times, that is (by supposition), to the velocities, and so to the 
spaces described. Take the sums of the analogous quantities, and the areas 
Bkq, Elr, Ems, But, (fee., will be analogous to the whole spaces described ; 
and also the areas AB<?K, ABrL, ABsM, AB^N, (fee., to the times. There 
fore the body, in descending, will in any time ABrL describe the space Blr, 
and in the time Lr^N the space rlnt. Q,.E.D. And the like demonstra 
tion holds in ascending motion. 

COROL. 1. Therefore the greatest velocity that the body can acquire by 
falling is to the velocity acquired in any given time as the iven force ol 
gravity which perpetually acts upon it to the resisting force which opposes 
it at the end of that time. 



854 



THE MATHEMATICAL PRINCIPLES 



[BOOK IL 



COROL. 2. But the time being augmented in an arithmetical progression, 
the sum of that greatest velocity and the velocity in the ascent, and also 
their difference in the descent, decreases in a geometrical progression. 

COROL. 3. Also the differences of the spaces, which are described in equal 
differences of the times, decrease in the same geometrical progression. 

COROL. 4. The space described by the body is the difference of two 
spaces, whereof one is as the time taken from the beginning of the descent, 
and the other as the velocity; which [spaces] also at the beginning of the 
descent are equal among themselves. 

PROPOSITION IV. PROBLEM II. 

Supposing the force of gravity in any similar medium to be uniform, 
and to tend perpendicularly to the plane of the horizon ; to define the 
motion of a projectile therein, which suffers resistance proportional to 
its velocity. 

Let the projectile go from any place D in 
the direction of any right line DP, and let 
its velocity at the beginning of the motion 
be expounded by the length DP. From the 
point P let fall the perpendicular PC on the 
horizontal line DC, and cut DC in A, so 
that DA may be to AC as the resistance 
of the medium arising from the motion up 
wards at the beginning to the force of grav 
ity; or (which comes to the same) so that 
t ie rectangle under DA and DP may be to 
that under AC and CP as the whole resist 
ance at the beginning of the motion to the 
force of gravity. With the asymptotes 
DC, CP describe any hyperbola GTBS cut 
ting the perpendiculars DG, AB in G and 
B ; complete the parallelogram DGKC, and 
let its side GK cut AB in Q,. Take 




N in the same ratio to QB as DC is in to CP ; and from any point R of the 
right line DC erect RT perpendicular to it, meeting the hy] erbola in T, 
and the right lines EH, GK, DP in I, t, and V ; in that perpendicular 



take Vr equal to ~- , or which is the same thing, take Rr equal to 

(""""PIT? 

^ T ; and the projectile in the time DRTG will arrive at the point r 

describing the curve line DraF, the locus of the point r ; thence it will 
come to its greatest height a in the perpendicular AB j and afterwards 



SEC. 1.J OF NATURAL PHILOSOPHY. 255 

ever approach to the asymptote PC. And its velocity in any pjint r will 
be as the tangent rL to the curve. Q.E.I. 

For N is to Q,B as DC to CP or DR to RV, and therefore RV is equal to 

PR X QB , -..". "v v DRXQB-/GT 

^r - , and R/ (that is, RV Vr, or - -- ^ --- ) is equal to 

DR X Ap RDGT 

-- ~ --- . JNow let the time be expounded by the area 

RDGT and (by Laws, Cor. 2), distinguish the motion of the body into 
two others, one of ascent, the other lateral. And since the resistance is as 
the motion, let that also be distinguished into two parts proportional and 
contrary to the parts of the motion : and therefore the length described by 
the lateral motion will be (by Prop. II, Book II) as the line DR, and the 
height (by Prop. Ill, Book II) as the area DR X AB RDGT, that is, 
as the line Rr. But in the very beginning of the motion the area RDGT 
is equal to the rectangle DR X AQ, and therefore that line Rr (or 
DRx AB 



that is, as CP to DC ; and therefore as the motion upwards to the motion 
lengthwise at the beginning. Since, therefore, Rr is always as the height, 
and DR always as the length, and Rr is to DR at the beginning as the 
height to the length, it follows, that Rr is always to DR as the height to 
the length ; and therefore that the body will move in the line DraF. which 
is the locus of the point r. QJE.D. 

DR X AB RDGT 

COR. 1. Therefore Rr is equal to -- ^ ------ ^ - . and therefore 

if RT be produced to X so that RX may be equal to -- ^ -- ; that is, 

if the parallelogram ACPY be completed, and DY cutting CP in Z be 
drawn, and RT be produced till it meets DY in X ; Xr will be equal to 

RDGT 

^ , and therefore proportional to the time. 

COR. 2. Whence if innumerable lines CR, or, which is the same, innu 
merable lines ZX, be taken in a geometrical progression, there will be as 
many lines Xr in an arithmetical progression. And hence the curve DraF 
is easily delineated by the table of logarithms. 

COR. 3. If a parabola be constructed to the vertex D, and the diameter 
DG produced downwards, and its latus rectum is to 2 DP as the whole 
resistance at the beginning of the notion to the gravitating force, the ve 
locity with which the body ought *o go from the place D, in the direction 
of the right line DP, so as in an uniform resisting medium to describe the 
curve DraF, will be the same as that with which it ought to go from the 
same place D in the direction of the same right line DP, so as to describe 



256 



THE MATHEMATICAL PRINCIPLES ~ 



[BOOK II 




I a parabola in a non-resisting medium. For 
the latus rectum of this parabola, at the very 

DV 2 

beginning of the motion, is -y- , andVris 

tGT DR x T* 

-~JT- or ^T . But a right line, which, 

if drawn, would touch the hyperbola GTS in 
K G, is parallel to DK, and therefore T* is 



CKX DR 



c 



QBx DC 



^ , and N is ~pp Ahd there- 



DC 

DR 2 X CK x CP 
fore Vr is equal to 2DC 2 X QlT~ ; *^ at * S (Because D ^ an< * ^)C, DV 

DV 2 x CK ~x OP 

and DP are proportionals), to ^T5 Fcrr J an <* tne ^ atus reeturn 



DV 2 



- comes out - 



2DP 2 X QB 



are proportional), 



CK X CP 
2DP 2 X DA 



AC X CP 

CP X AC ; that is, as the resistance to the gravity. 



( because 



and therefore is to 2DP as DP X DA to 



Q.E.D. 



COR. 4. Hence if a body be projected from 
any place D with a given velocity, in the 
direction of a right line DP given by posi 
tion, and the resistance of the medium, at 
the beginning of the motion, be given, the 
curve DraF, which that body will describe, 
may be found. For the velocity being 
given, the latus rectum of the parabola is 
given, as is well known. And taking 2DP 
to that latus rectum, as the force of gravity 
to the resisting force, DP is also given. 
Then cutting DC in A, so that GP X AC 
may be to DP X DA in the same ratio of 
the gravity to the resistance, the point A 
will be given. And hence the curve DraF 
is also given. 

COR. 5. And, on the contrary, if the 
curve DraF be given, there will be given 
x>th the velocity of the body and the resistance of the medium in each of 
the places r. For the ratio of CP X AC to DP X DA being given, there 
is given both the resistance of the medium at the beginning of the motion 
and the latus rectum of the parabola ; and thence the velocity at the be 
ginning of the motion is given also. Then from the length of the tangent 





SEC. I.] OF NATURAL PHILOSOPHY. 257 

L there is given both the velocity proportional to it, and the resistance 
proportional to the velocity in any place r. 

COR. 6. But since the length 2DP is to the latus rectum of the para 
bola as the gravity to the resistance in D ; and, from the velocity aug 
mented, the resistance is ti gmented in the same ratio, but the latus rectum 
of the parabola is augmented in the duplicate of that ratio, it is plain thot 
the length 2DP is augmented in that simple ratio only ; and is therefore 
always proportional to the velocity ; nor will it be augmented or dimin 
ished by the change of the angle CDP, unless the velocity be also changed. 

COR. 7. Hence appears the method of deter 
mining the curve DraF nearly from the phe- 
nomena, and thence collecting the resistance and 
velocity with which the body is projected. Let 
two similar and equal bodies be projected with 
the same velocity, from the place D, in differ 
ent angles CDP, CDp ; and let the places F, 
f. where they fall upon the horizontal plane 
DC, be known. Then taking any length for D */ F 

DP or Dp suppose the resistance in D to be to 
the suavity in any ratio whatsoever, and let that 

ratio be expounded by any length SM. Then, , _ 

by computation, from that assumed length DP, ^x 

find the lengths DF, D/; and from the ratio 

F/ 

-p^, found by calculation, subduct the same ratio as found by experiment ; 

and let the cKfference be expounded by the perpendicular MN. Repeat the 
same a second and a third time, by assuming always a new ratio SM of the 
resistance to the gravity, and collecting a new difference MN. Draw the 
affirmative differences on one side of the right line SM, and the negative 
on the other side ; and through the points N, N, N, draw a regular curve 
NNN. cutting the right line SMMM in X, and SX will be the true ratio 
of the resistance to the gravity, which was to be found. From this ratio 
the length DF is to be collected by calculation ; and a length, which is to 
the assumed length DP as the length DF known by experiment to the 
length DF just now found, will be the true length DP. This being known, 
you will have both the curve line DraF which the body describes, and also 
the velocity and resistance of the body in each place. 

SCHOLIUM. 

But, yet, that the resistance of bodies is in the ratio of the velocity, is more 
a mathematical hypothesis than a physical one. In mediums void of all te 
nacity, the resistances made to bodies are in the duplicate ratio of the ve 
locities. For by the action of a swifter body, a greater motion in propor- 

17 



THE MATHEMATICAL PRINCIPLES [BoOK IL 

tion to a greater velocity is communicated to the same quantity of the 
medium in a less time ; and in an equal time, by reason of a greater quan 
tity of the disturbed medium, a motion is communicated in the duplicate 
ratio greater ; and the resistance (by Law II and III) is as the motion 
communicated. Let us, therefore, see what motions arise from this law of 
resistance. 



SECTION II. 

If the motion of bodies that are resisted in tfie duplicate ratio of their 

velocities. 

PROPOSITION V. THEOREM III. 

Ff a body is resisted in the duplicate ratio of its velocity, and moves by 
its innate force only through a similar medium; and the times be 
taken in a geometrical progression, proceeding from less to greater 
terms : I say, that the velocities at the beginning of each of the times 
are in the same geometrical progression inversely ; and that the spaces 
are equal, which are described in each of the times. 
For since the resistance of the medium is proportional to the square of 
the velocity, and the decrement of the velocity is proportional to the resist 
ance : if the time be divided into innumerable equal particles, the squares of 
the velocities at the beginning of each of the times will be proportional to 
the differences of the same velocities. Let those particles of time be AK, 
KL, LM, &c., taken in the right line CD; and 
erect the perpendiculars AB, Kk, L/, Mm, &c., 
meeting the hyperbola BklmG, described with the 
centre C, and the rectangular asymptotes CD, CH. 
in B, kj I, m, (fee. ; then AB will be to Kk as CK 
to CA, and, by division, AB Kk to Kk as AK 

C ARIMT to ^ A> an(1 alternate ^ AB ^ C to AK as Kk 

to CA ; and therefore as AB X Kk to AB X CA. 

Therefore since AK and AB X CA are given,* AB Kk will be as AB 
X Kk ; and, lastly, when AB and KA* coincide, as AB 2 . And, by the like 
reasoning, KAr-U, J J-M/??, (fee., will be as Kk 2 . LI 2 , (fee. Therefore the 
squares of the lines AB, KA", L/, Mm, (fee., are as their differences ; and, 
therefore, since the squares of the velocities were shewn above to be as their 
differences, the progression of both will be alike. This being demonstrated 
it follows also that the areas described by these lines are in a like progres 
sion with the spaces described by these velocities. Therefore if the velo 
city at the beginning of the first time AK be expounded by the line AB, 




SEC. II.] OF NATURAL PHILOSOPHY. 25 C J 

and the velocity at the beginning of the second time KL by the line K& 
and the length described in the hrst time by the area AKA*B, all the fol 
lowing velocities will be expounded by the following lines \J. Mm, .fee. 
and the lengths described, by the areas K/, I mi. &c. And, by compo 
sition, if the whole time be expounded by AM, the sum of its parts, the 
whole length described will be expounded by AM/ftB the sum of its parts. 
Now conceive the time AM to be divided into the parts AK, KL, LM, (fee 
so that CA, CK, CL, CM, (fee. may be in a geometrical progression ; and 
those parts will be in the same progression, and the velocities AB, K/r, 
L/, Mm, (fee., will be in the same progression inversely, and the spaces de 
scribed Ak, K/, Lw, (fee., will be equal. Q,.E.D. 

COR. 1. Hence it appears, that if the time be expounded by any part 
AD of the asymptote, and the velocity in the beginning of the time by the 
ordinate AB, the velocity at the end of the time will be expounded by the 
ordinate DG ; and the whole space described by the adjacent hyperbolic 
area ABGD ; and the space which any body can describe in the same time 
AD, with the first velocity AB, in a non-resisting medium, by the rectan 
gle AB X AD. 

COR 2. Hence the space described in a resisting medium is given, by 
taking it to the space described with the uniform velocity AB in a non- 
resisting medium, as the hyperbolic area ABGD to the rectangle AB X AD. 

COR. 3. The resistance of the medium is also given, by making it equal, 
in the very beginning of the motion, to an uniform centripetal force, which 
could generate, in a body falling through a non-resisting medium, the ve 
locity AB in the time AC. For if BT be drawn touching the hyperbola 
in B. and meeting the asymptote in T, the right line AT will be equal to 
AC, and will express the time in which the first resistance, uniformly con 
tinned, may take away the whole velocity AB 

COR. 4. And thence is also given the proportion of this resistance to the 
force of gravity, or ay other given centripetal force. 

COR. 5. And, vice versa, if there is given the proportion of the resist- 
; nee to any given centripetal force, the time AC is also given, in which c 
centripetal force equal to the resistance may generate any velocity as AB ; 
and thence is given the point B. through which the hyperbola, having CH 
CD for its asymptotes, is to be described : as also the space ABGD, which a 
body, by beginning its motion with that velocity AB, can describe in any 
time AD. in a similar resisting medium. 

PROPOSITION VI. THEOREM lV r c 

Homogeneous and equal spherical bodies, opposed hy resistances that are 
in the duplicate ratio of the velocities, and moving on by their innate 
force only, will, in times which are reciprocally as the velocities at thr. 



260 THE MATHEMATICAL PRINCIPLES |BOOK II, 



v 




beg-in fiing, describe equal spaces, and lose parts of their velocities pro 
portional to the wholes. 

To the rectangular asymptotes CD, CH de 
scribe any hyperbola B6Ee, cutting the perpen 
diculars AB, rib, DE, de in B, b, E, e; let the 
initial velocities be expounded by the perpendicu 
lars AB, DE, and the times by the lines Aa, Drf. 
Therefore as Aa is to l)d, so (by the hypothesis) 
. is DE to AB, and so (from the nature of the hy- 
C "^ perbola) is CA to CD ; and, by composition, so is 

Crt to Cd. Therefore the areas ABba, DEerf, that is, the spaces described, 
are equal among themselves, and the first velocities AB, DE are propor 
tional to the last ab, de ; and therefore, by division, proportional to the 
parts of the velocities lost, AB ab, DE de. Q.E.D. 

PROPOSITION VII. THEOREM V. 

If spherical bodies are resisted in the duplicate ratio of their velocities, 
in times which are as the first motions directly, and the first resist 
ances inversely, they will lose parts of their motions proportional to the 
wholes, and will describe spaces proportional to those times and the 
first velocities conjunct It/. 

For the parts of the motions lost are as the resistances and times con 
junctly. Therefore, that those parts may be proportional to the wholes, 
the resistance and time conjunctly ought to be as the motion. Therefore the 
time will be as the motion directly and the resistance inversely. Where 
fore the particles of the times being taken in that ratio, the bodies will 
always loso parts of their motions proportional to the wholes, and there 
fore will retain velocities always proportional to their first velocities. 
And because of the given ratio of the velocities, they will always describe 
spaces which are as the first velocities and the times conjunctly. Q.E.D. 
COR. 1. Therefore if bodies equally swift are resisted in a duplicate ra 
tio of their diameters, homogeneous globes moving with any velocities 
whatsoever, by describing spaces proportional to their diameters, will lose 
parts of their motions proportional to the wholes. For the motion of each 
o-lobe will be as its velocity and mass conjunctly, that is, as the velocity 
and the cube of its diameter ; the resistance (by supposition) will be as the 
square of the diameter and the square of the velocity conjunctly ; and the 
time (by this proposition) is in the former ratio directly, and in the latter 
inversely, that is, as the diameter directly and the velocity inversely ; and 
therefore the space, which is proportional to the time and velocity is as 
the diameter. 

COR. 2. If bodies equally swift are resisted in a sesquiplicate ratio of 
their diameters, homogeneous globes, moving with any velocities whatso- 



SEC. 1L] OF NATURAL PHILOSOPHY. 261 

ever, by describing spaces that are in a sesquiplicate ratio of the diameters, 
will lose parts of their motions proportional to the wholes. 

COR. 3. And universally, if equally swift bodies are resisted in the ratio 
of any power of the diameters, the spaces, in which homogeneous globes, 
moving with any velocity whatsoever, will lose parts of their motions pro 
portional to the wholes, will be as the cubes of the diameters applied to 
that power. Let those diameters be D and E : and if the resistances, where 
the velocities are supposed equal, are as T) n and E" ; the spaces in which 
the globes, moving with any velocities whatsoever, will lose parts of their 
motions proportional to the wholes, will be as D 3 n and E 3 n . And 
therefore homogeneous globes, in describing spaces proportional to D 3 n 
and E 3 n , will retain their velocities in the same ratio to one another as 
at the beginning. 

COR. 4. Now if the globes are not homogeneous, the space described by 
the denser globe must be augmented in the ratio of the density. For the 
motion, with an equal velocity, is greater in the ratio of the density, and 
the time (by this Prop.) is augmented in the ratio of motion directly, and 
the space described in the ratio of the time. 

COR. 5. And if the globes move in different mediums, the space, in a 
medium which, cccteris paribus, resists the most, must be diminished in the 
ratio of the greater resistance. For the time (by this Prop.) will be di 
minished in the ratio of the augmented resistance, and the space in the ra 
tio of the time. 

LEMMA II. 

The moment of any genitum is equal to the moments of each of the gen- 
eratinrr sides drawn into the indices of the powers of those sides, and 
into their co-efficients continually. 

I call any quantity a genitum which is not made by addition or sub- 
duction of divers parts, but is generated or produced in arithmetic by the 
multiplication, division, or extraction of the root of any terms whatsoever : 
in geometry by the invention of contents and sides, or of the extremes and 
means of proportionals. Quantities of this kind are products, quotients, 
roots, rectangles, squares, cubes, square and cubic sides, and the like. 
These quantities I here consider as variable and indetermined, and increas 
ing or decreasing, as it were, by a perpetual motion or flux ; and I under 
stand their momentaneous increments or decrements by the name of mo 
ments ; so that the increments may be esteemed as added or affirmative 
moments ; and the decrements as subducted or negative ones. But take 
care not to look upon finite particles as such. Finite particles are not 
moments, but the very quantities generated by the moments. We are to 
conceive them as the just nascent principles of finite magnitudes. Nor do 
we in this Lemma regard the magnitude of the moments, but their firsf 



262 THE MATHEMATICAL PRINCIPLES [BoOK 11 

proportion, as nascent. It will be the same thing, if, instead of moments, 
we use either the velocities of the increments and decrements (which may 
also be called the motions, mutations, and fluxions of quantities), or any 
finite quantities proportional to those velocities. The co-efficient of any 
generating side is the quantity which arises by applying the genitum to 
ihat side. 

Wherefore the sense of the Lemma is, that if the moments of any quan 
tities A, B, C, &c., increasing or decreasing by a perpetual flux, or the 
velocities of the mutations which are proportional to them, be called a, 6, 
r, (fee., the moment or mutation of the generated rectangle AB will be B 
-h b A ; the moment of the generated content ABC will be aBC -f b AC 4 

-1 -2. .1 

cAB; and the moments of the generated powers A 2 . A 3 , A 4 , A 2 , A 2 . A 3 , 
A*, A , A 2 , A * will be 2aA, 3a A 2 , 4aA 3 , |A *, fA* 

11 3 

i A s , |/iA 3 , a A 2 , 2aA 3 , aA 2 respectively; and 
in general, that the moment of any power A^, will be ^ aA n -^. Also, 
that the moment of the generated quantity A 2 B will be 2aAB 4- bA~ ; the 
moment of the generated quantity A 3 B 4 C 2 will be 3A 2 B 4 C 2 + 4/>A 3 

A 3 
B 3 C 2 4-2cA 3 B C; and the moment of the generated quantity or 

A B 2 will be 3aA 2 B 2 2bA 3 B 3 ; and so on. The Lemma is 
thus demonstrated. 

CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, as 
yet, there wanted of the sides A and B half their moments \a and \b, was 
A \a into B \b, or AB a B \b A + \ab ; but as soon as the 
sides A and B are augmented by the other half moments, the rectangle be 
comes A 4- 4-a into B 4- \b, or AB -f ^a B 4- \b A -f \ab. From this 
rectangle subduct the former rectangle, and there will remain the exces.? 
aE -f bA. Therefore with the whole increments a and b of the sides, tin 
increment aB + f>A of the rectangle is generated. Q.K.D. 

CASE 2. Suppose AB always equal to G, and then the moment of the 
content ABC or GC (by Case 1) will be^C + cG, that is (putting AB and 
aB + bA for G and *), aBC -h bAC 4- cAB. And the reasoning is the 
same for contents under ever so many sides. Q.E.D. 

CASE 3. Suppose the sides A, B, and C, to be always equal among them 
selves; and the moment B + />A, of A 2 , that is, of the rectangle AB, 
will be 2aA ; and the moment aBC + bAC + cAB of A 3 , that is, of the 
content ABC, will be 3aA 2 . And by the same reasoning the moment of 
any power A n is naA n . Q.E.D 

CASE 4. Therefore since -7 into A is 1, the moment of -r- drawn into 
A A 



SEC. 11.] OF NATURAL PHILOSOPHY. 263 

A, together with drawn into a. will be the moment of 1, that is, nothing. 
A 

Therefore the moment of -r, or of A , is -r . And generally since 

A .A 

T- into A n is I, the moment of drawn into A n together with into 
A n A. A n 

naA" ! will be nothing. And, therefore, the moment of -r- or A n 

A 

will be T^~7- Q-E.D. 

V . t . i 

CASE 5. And since A 2 into A 2 is A, the moment of A 1 drawn into 2 A 2 

will be a (by Case 3) ; and, therefore, the moment of A 7 will be n~r~r or 

^A-j 

#A . And, generally, putting A~^ equal to B, then A m will be equal 
to B n , and therefore maA m ! equal to nbB n , and ma A equal to 

?tbB , or tib A ^ 5 an< i therefore ri a A ^~ is equal to &, that is, equal 

to the moment of A^. Q.E.D. 

CASE 6. Therefore the moment of any generated quantity A m B n is the 
moment of A m drawn into B n , together with the moment of B n drawn into 
A", that is, maA m B" -f- nbB n ! A m ; and that whether the indices 
in arid n of the powers be whole numbers or fractions, affirmative or neg 
ative. And the reasoning is the same for contents under more powers. 
Q.E.D. 

COR. 1. Henoe in quantities continually proportional, if one term is 
given, the moments of the rest of the terms will be as the same terms mul 
tiplied by the number of intervals between them and the given term. Let 
A, B, C, D ; E, F, be continually proportional ; then if the term C is given, 
the moments of the rest of the terms will be among themselves as 2A, 
B ? D, 2E, 3F. 

COR. 2. And if in four proportionals the two means are given, the mo 
ments of the extremes will be as those extremes. The same is to be un 
derstood of the sides of any given rectangle. 

COR. 3. And if the sum or difference of two squares is given, the mo 
ments of the sides will be reciprocally as the sides. 

SCHOLIUM. 

In a letter of mine to Mr. /. Collins, dated December 10, 1672, having 
described a method of tangents, which I suspected to be the same with 
Slusius*s method, which at that time wag not made public, I subjoined these 
words This is one particular, or rather a Corollary, of a general nte 



264 THE MATHEMATICAL PRINCIPLES [BjOK II. 

thod, which extends itself, without any troublesome calculation, not ojdy 
to the drawing of tangents to any curve lines, whether geometrical or 
mechanical, or any how respecting right lines or other cnrves, but also 
to the resolving other abstrnser kinds of problems about the crookedness, 
areas, lengths, centres of gravity of curves, &c. ; nor is it (as Hudd^ri s 
method de Maximis & Minimia) limited to equations which are free from 
surd quantities. This method I have interwoven with that other oj 
working in equations, by reducing them to infinite serie?. So far that 
letter. And these last words relate to a treatise I composed on that sub 
ject in the year 1671. The foundation of that general method is contain 
ed in the preceding Lemma. 

PROPOSITION VIII. THEOREM VI. 

If a body in an uniform medium, being uniformly acted upon by the force 
of gravity, ascends or descends in a right line ; and the whole space 
described be distinguished into equal parts, and in the beginning of 
each of the parts (by adding or subducting the resisting force of the 
medium to or from the force of gravity, when the body ascends or de 
scends] yon collect the absolute forces ; I say, that those absolute forces 
ire in a geometrical progression. 

For let the force of gravity be expounded by the 
given line AC ; the force of resistance by the indefi 
nite line AK ; the absolute force in the descent of the 
body by the difference KC : the velocity of the I tody 
<^LKJL&i>F/ by a line AP, which shall be a mean proportional be 
tween AK and AC, and therefore in a subduplicate ratio of the resistance; 
the increment of the resistance made in a given particle of time by the li- 
neola KL, and the contemporaneous increment of the velocity by the li- 
neola PQ ; and with the centre C, and rectangular asymptotes CA, CH, 
describe any hyperbola BNS meeting the erected perpendiculars AB, KN, 
LO in B, N and O. Because AK is as AP 2 , the moment KL of the one will 
be as the moment 2APQ of the other, that is, as AP X KC ; for the in 
crement PQ of the velocity is (by Law II) proportional to the generating 
force KC. Let the ratio of KL be compounded with the ratio KN, and 
the rectangle KL X KN will become as AP X KC X KN ; that is (because 
the rectangle KC X KN is given), as AP. But the ultimate ratio of the 
hyperbolic area KNOL to the rectangle KL X KN becomes, when the 
points K and L coincide, the ratio of equality. Therefore that hyperbolic 
evanescent area is as AP. Therefore the whole hyperbolic area ABOL 
is composed of particles KNOL which are always proportional to the 
velocity AP; and therefore is itself proportional to the space described 
with that velocity. Let ,that area be now divided into equal parts 



SEC. IJ.J 



OF NATURAL PHILOSOPHY. 



265 



as ABMI, IMNK, KNOL, (fee., and the absolute forces AC, 1C, KC, LC, 

(fee., will be in a geometrical progression. Q,.E.D. And by a like rea 
soning, in the ascent of the body, taking, on the contrary side of the point 
A, the equal areas AB?m, i/nnk, knol, (fee., it will appear that the absolute 
forces AC. iG, kC, 1C, (fee., are continually proportional. Therefore if all 
the spaces in the ascent and descent are taken equal, all the absolute forces 
1C, kC, iC, AC, 1C, KC, LC, (fee., will be continually proportional. Q,.E.D. 

COR. 1. Hence if the space described be expounded by the hyperbolic 
area ABNK, the force of gravity, the velocity of the body, and the resist 
ance of the medium, may be expounded by the lines AC, AP, and AK re 
spectively and vice versa. 

COR. 2. And the greatest velocity which the body can ever acquire in 
an infinite descent will be expounded by the line AC. 

COR. 3. Therefore if the resistance of the medium answering to any 
given velocity be known, the greatest velocity will be found, by taking it 
to that given velocity in a ratio subduplicate of the ratio which the force 
of gravity bears to that known resistance of the medium. 

PROPOSITION IX. THEOREM VII. 

Supposing ivhat is above demonstrated, I say, that if the tangents of t-he 
angles of the sector of a circle, and of an hyperbola, be taken propor 
tional to the velocities, the radius being of a fit magnitude, all the time 
of the ascent to the highest place icill be as the sector of the circle, and 
all the time of descending from the highest place as the sector of t/ie 
hyperbola. 

To the right line AC, which ex 
presses the force of gravity, let AD 
drawn perpendicular and equal. From 
the centre D with the semi-diameter 
AD describe as well the cmadrant A^E 

-t 

of a circle, as the rectangular hyper 
bola AVZ, whose axis is AK, principal 
vertex A, and asymptote DC. Let Dp, 
DP be drawn ; and the circular sector 
AtD will be as all the time of the as 
cent to the highest place ; and the hy 
perbolic sector ATD as all the time of descent from the highest place; ii 
BO be that the tangents Ap, AP of those sectors be as the velocities. 

CASE 1. Draw Dvq cutting off the moments or least particles tDv and 
^ ?, described in the same time, of the sector ADt and of the triangle 
AD/?. Since those particles (because of the common angle D) are in a du 
plicate ratio of the sides, the particle tDv will be as -^-^- , that is 




266 



THE MATHEMATICAL PRINCIPLES 



[BOOK li. 



(because tD is given), as ^f. But joD 8 is AD 3 + Ap 2 , that is, AD 2 -h 
AD X AA-, or AD X Gk ; and (/Dp is 1 AD X pq. Therefore tDv, the 

BO 

particle of the sector, is as ^ , ; that is, as the least decrement pq of the 

velocity directly, and the force Gk which diminishes the velocity, inversely ; 
and therefore as the particle of time answering to the decrement of the ve 
locity. And, by composition, the sum of all the particles tDv in the sector 
AD/ will be as the sum of the particles of time answering to each of the 
lost particles pq of the decreasing velocity Ap, till that velocity, being di 
minished into nothing, vanishes; that is, the whole sector AD/ is as the 
whole time of ascent to the highest place. Q.E.D. 

CASE 2. Draw DQV cutting off the least particles TDV and PDQ of 
the sector DAV, and of the triangle DAQ ; and these particles will be to 
each other as DT 2 to DP 2 , that is (if TX and AP are parallel), as DX 2 
to DA 2 or TX 2 to AP 2 ; and, by division, as DX 2 TX 2 to DA 2 - 
AP 2 . But. from the nature of the hyperbola, DX 2 TX 2 is AD 2 ; and, by 
the supposition, AP 2 is AD X AK. Therefore the particles are to each 
other as AD 2 to AD 2 AD X AK ; that is, as AD to AD AK or AC 

; and 



to CK : and therefore the particle TDV of the sector is - 

PQ 



therefore (because AC and AD are given) as 



CK 

that is, as the increment 



of the velocity directly, and as the force generating the increment inverse 
ly ; and therefore as the particle of the time answering to the increment. 
And, by composition, the sum of the particles of time, in which all the par 
ticles PQ of the velocity A I 3 are generated, will be as the sum of the par 
ticles of the sector ATI) ; that is, the whole time will be as the whole 
sector. Q.E.D. 

COR. 1. Hence if AB be equal to a 
fourth part of AC, the space which a body 
will describe by falling in any time will 
be to the space which the body could de 
scribe, by moving uniform]} on in the 
same time with its greatest velocity 
AC, as the area ABNK, which ex 
presses the space described in falling to 
the area ATD, which expresses the 
time. For since AC is to AP as AP 
_ to AK, then (by Cor. 1, Lem. II, of this 

Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence 
LK is to ^PQ as AP to JAC or AB ; and KN is to AC or AD as AB tc 




. II.] OF NATURAL PHILOSOPHY. 267 

UK ; and therefore, ex cequo, LKNO to DPQ, as AP to CK. But DPQ 
was to DTV as CK to AC. Therefore, ex aquo, LKNO is to DTV r,? 
AP to AC ; that is, as the velocity of the falling body to the greatest 
velocity which the body by falling can acquire. Since, therefore, the 
moments LKNO and DTV of the areas ABNK and ATD are as the ve 
locities, all the parts of those areas generated in the same time will be as 
the spaces described in the same time ; and therefore the whole areas ABNK 
and ADT, generated from the beginning, will be as the whole spaces de 
scribed from the beginning of the descent. Q.E.D. 

COR. 2. The same is true also of the space described in the ascent. 
That is to say, that all that space is to the space described in the same 
time, with the uniform velocity AC, as the area ABttk is to the sector ADt. 

COR. 3. The velocity of the body, falling in the time ATD, is to the 
velocity which it would acquire in the same time in a non-resisting space, 
as the triangle APD to the hyperbolic sector ATD. For the velocity in 
a non-resisting medium Avould be as the time ATD, and in a resisting me 
dium is as AP, that is, as the triangle APD. And those velocities, at the 
beginning of the descent, are equal among themselves, as well as those 
areas ATD, APD. 

COR. 4. By the same argument, the velocity in the ascent is to the ve 
locity with which the body in the same time, in a non-resisting space, would 
lose all its motion of ascent, as the triangle ApD to the circular sector 
AtD ; or as the right line Ap to the arc At. 

COR. 5. Therefore the time in which a body, by falling in a resisting 
medium, would acquire the velocity AP, is to the time in which it would 
acquire its greatest velocity AC, by falling in a non-resisting space, as the 
sector ADT to the triangle ADC : and the time in which it would lose its 
velocity Ap, by ascending in a resisting medium, is to the time in which 
it would lose the same velocity by ascending in a non-resisting space, as 
the arc At to its tangent Ap. 

COR. 6. Hence from the given time there is given the space described in 
the ascent or descent. For the greatest velocity of a body descending in 
wfinitum is given (by Corol. 2 and 3, Theor. VI, of this Book) ; and thence 
the time is given in which a body would acquire that velocity by falling 
in a non-resisting space. And taking the sector ADT or ADt to the tri 
angle ADC in the ratio of the given time to the time just now found, 
there will be given both the velocity AP or Ap, and the area ABNK or 
AB//&, which is to the sector ADT, or AD/, as the space sought to the 
space which would, in the given time, be uniformly described with that 
greatest velocity found just before. 

COR. 7. And by going backward, from the given space of ascent or de 
scent AB?A: or ABNK, there will be given the time AD* or ADT. 



268 



THE MATHEMATICAL PRINCIPLES 



[BOOK ii 




PROPOSITION X. PROBLEM III. 

Suppose the uniform force of gravity to tend directly to the plane of the 
horizon, and the resistance to be as the density of the medium and the 
square of the velocity coiijunctly : it is proposed to find the density of 
the medium in each place, which shall make the body move in any 
given carve line ; the velocity of the body and the resistance of the 
medium in each place. 

Let PQ be a plane perpendicular to 
the plane of the scheme itself; PFHQ 
a curve line meeting that plane in the 
points P and Q ; G, H, I, K four 
places of the body going on in this 
\ curve from F to Q ; and GB, HC, ID, 
KE four parallel ordinates let fall 

p A. 33 c^D E Q from these points to the horizon, and 

standing on the horizontal line PQ at the points B, C, D, E ; and let the 
distances BC, CD, DE, of the ordinates be equal among themselves. From 
the points G and H let the right lines GL, HN, be drawn touching the 
curve in G and H, and meeting the ordinates CH, DI, produced upwards, 
in L and N : and complete the parallelogram HCDM. And the times in 
which the body describes the- arcs GH, HI, will be in a subduplicate ratio 
of the altitudes LH, NI ; which the bodies would describe in those times, 
by falling from the tangents; and the velocities will be as the lengths de 
scribed GH, HI directly, and the times inversely. Let the times be ex- 

C*TT TTT 
pounded by T and t, and the velocities by =- and ---; and the decrement 

J_ L 

/-^TT TTT 

of the velocity produced in the time t will be expounded by -7^ . 

This decrement arises from the resistance which retards the body, and from 
the gravity which accelerates it. Gravity, in a falling body, which in its 
fall describes the space NI, produces a velocity with which it would be able 
to describe twice that space in the same time, as Galileo has demonstrated ; 

2NI 



that is, the velocity 



: but if the body describes the arc HI, it augments 
MIxNl 



HI 



; and therefore generates 



that arc only by the length HI HN or 

only the velocity iff- Let this velocity be added to the before- 

t X H.JL 

mentioned decrement, and we shall have the decrement of the velocity 

GH HI SMI X Nl 

arising from the resistance alone, that is, -^ : h 



T 



SEC. II.J 



OF NATURAL PHILOSOPHY. 



269 



2NI. 




Therefore since, in the same time, the action of gravity generates, in a fall 
ing body, the velocity , the resistance will be to the gravity as 7^ 

HI 2MI X NI 2NI t X GH 2MI X NI 

+ TTT- to or as ^ HI -f 

Now for the abscissas CB, CD, 
CE, put o, o, 2o. For the ordinate 
CH put P ; and for MI put any series 
Qo + Ro 2 + So 3 +, &c. And all 
the terms of the series after the first, 
that is, Ro 2 + So 3 +, (fee., will be 
NI ; and the ordinates DI, EK, and 

BGwill be P QoRo 2 So 3 p A B c T> E 

(fee., P 2Qo 4Ro 2 SSo 3 , (fee., and P -\- Qo Ro 2 + So 3 , 
(fee., respectively. And by squaring the differences of the ordinates BG 
CH and CH DI, and to the squares thence produced adding the squares 
of BC and CD themselves, you will have oo -f- QQoo 2QRo 3 +, (fee., 
and oo -f QQoo -f 2QRo 3 +, (fee., the squares of the arcs GH, HI ; whose 

QRoo QRoo 

roots o y - , and o </! 4- QQ 4- are the 

1 + QQ v/1 + QQ s/1 -f QQ 

arcs GH and HI. Moreover, if from the ordinate CH there be subducted 
half the sum of the ordinates BG and DI, and from the ordinate DI there 
be subducted half the sum of the ordinates CH and EK, there will remain 
Roo and Roo + 3So 3 , the versed sines of the arcs GI and HK. And these 
are proportional to the lineolae LH and NI, and therefore in the duplicate 

ratio of the infinitely small times T and t : and thence the ratio ~, is ^ 



R + SSo R -f 

^ or 



So , t X GH TTT 2MI X NI , 

: and ^ HI H TTT , by substituting 



R T HI 

the values of , GH, HI, MI and NI just found, becomes -^- 

J- /w-Lt/ 



I + QQ. Arid since 2NI is 2Roo, the resistance will be now to the 

OO 

gravity as -- TT Q to 2Roo > that is > as 3S r to 4RR. 



And the velocity will be such, that a body going off therewith from any 
place H, in the direction of the tangent HN, would describe, in vacuo, a 

parabola, whose diameter is HC, and its latus rectum NT or -- ^ ---- . 

And the resistance is as the density of the medium and the square of 
the velocity conjunctly ; and therefore the density of the medium is as the 
resistance directly, and the square of the velocity inversely ; that is, as 



270 THE MATHEMATICAL PRINCIPLES [BOOK II. 



QQ __ 



4RR 

Q.E.I. 

COR. 1. If the tangent HN be produced both ways, so as to meet any 

HT 
ordinatc AF in T - will be equal to V /T+ QQ; an d therefore in what 



has gone before may be put for ^ \ -\- QQ. By this means the resistance 
will be to the gravity as 3S X HT to 4RR X AC ; the velocity will be a* 

r-pj --^, and the density of the medium will be as - TT -n. 
AO -v/ i Jti X H 1 

COR. 2. And hence, if the curve line PFHQ be denned by the relation 
between the base or abscissa AC and the ordinate CH, as is usual, and the 
value of the ordinate be resolved into a converging series, the Problem 
will be expeditiously solved by the first terms of the series ; as in the fol 
lowing examples. 

EXAMPLE 1. Let the line PFHQ, be a semi-circle described upon the 
diameter PQ, to find the density of the medium that shall make a projec 
tile move in that line. 

Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; and 
CD, o ; then DI 2 or AQ, 2 AD 2 = nn aa 2ao oo, or ec. 2ao 
oo ; and the root being extracted by our method, will give DI = e 
ao oo aaoo ao 3 a 3 o 3 
~e~~~2e 2e? ~~~ W ~2? &C * Here put nn f r ee + aa > and 

ao nnoo anno 3 

DI will become = e , &c. 

e 2e 3 2e 5 

Such series I distinguish into successive terms after this manner : I call 
that the first term in which the infinitely small quantity o is not found ; 
the second, in which that quantity is of one dimension only ; the third, in 
which it arises to two dimensions ; the fourth, in which it is of three ; and 
so ad infinitum. And the first term, which here is e, will always denote 
the length of the ordinate CH, standing at the beginning of the indefinite 

quantity o. The second term, which here is , will denote the difference 

between CH and DN ; that is, the lineola MN which is cut off by com 
pleting the parallelogram HC DM; and therefore always determines the 

CM? 

position of the tangent HN ; as, in this case, by taking MN to HM as 

G 

to o, or a to e. The third term, which here is -, will represent the li 
neola IN, which lies between the tangent and the curve ; and therefore 
determines the angle of contact IHN, or the curvature which the curve line 




SEC. II.] OF NATURAL PHILOSOPHY. 271 

has in H. If that lineola IN is of a finite magnitude, it will be expressed 
by the third term, together with those that follow in wfinitu:.:i. But if 
that lineola be diminished in. infini- 
tnm, the terms following become in 
finitely less than the third term, and 
therefore may be neglected. The 
fourth term determines the variation 
of the curvature ; the fifth, the varia 
tion of the variation ; and so on. 

Whence, by the way, appears no con-p~" ~K B~C~D~E~ 

temptible use of these series in the solution of problems that depend upon 
tangents, and the curvature of curves. 

ao 77/700 anno 3 

Now compare the series e ^ ^~ &c., with the 

e Ze 3 Ze* 

series P Qo - Roo So 3 &c., and for P, Q, II and S ? put e, -, ^-^ 

and ~ , and for ^ 1 + QQ put 1 H or - ; and the density oi 

2e 5 ee e 

the medium will come out as ; that is (because n is given), as - or 

lie e 

~Yj, that is, as that length of the tangent HT, which is terminated at the 
OH. 

semi-diameter AF standing perpendicularly on PQ : and the resistance 
will be to the gravity as 3a to 2>/, that is, as SAC to the diameter PQ of 
the circle; and the velocity will be as i/ CH. Therefore if the body goes 
from the place F, with a due velocity, in the direction of a line parallel to 
PQ, and the density of the medium in each of the places H is as the length 
of the tangent HT, and the resistance also in any place H is to the force 
of gravity as SAC to PQ, that body will describe the quadrant FHQ of a 
circle. Q.E.I. 

But if the same body should go-frorn the place P, in the direction of a 
line perpendicular to PQ, and should begin to move in an arc of the semi 
circle PFQ, we must take AC or a on the contrary side of the centre A ; 
and therefore its sign must be changed, and we must put a for + a. 

Then the density of the medium would come out as . But nature 

does not admit of a negative density, that is, a density which accelerates 
the motion of bodies; and therefore it cannot naturally come to pass that 
a body by ascending from P should describe the quadrant PF of a circle. 
To produce such an effect, a body ought to be accelerated by an impelling 
medium, and not impeded by a resisting one. 

EXAMPLE 2. Let the line PFQ be a parabola, having its axis AF per- 



272 



THE MATHEMATICAL PRINCIPLES 



[BOOK IL 




pendicular to the horizon PQ, to find the density of the medium, which 
will make a projectile move in that line. 

From the nature of the parabola, the rectangle PDQ, 
is equal to the rectangle under the ordinate DI and some 
given right line ; that is, if that right line be called b ; 
PC, a; PQ, c; CH, e; and CD, o; the rectangle a 
A. CD ~Q + o into c a o or ac aa 2ao -{-co oo, ia 

ac aa 

equal to the rectangle b into DI, and therefore DI is equal to -- 7 -- h 

c 2a oo c 2a 

-. o r. Now the second term -, o of this series is to be put 
b b b 

oo 

for Q,o, and the third term -r for Roo. But since there are no more 

terms, the co-efficient S of the fourth term will vanish ; and therefore the 

S 
ouantitv - , to which the density of the medium is proper- 





R v i 

tional, will be nothing. Therefore, where the medium is of no density, 

the projectile will move in a parabola ; as Galileo hath heretofore demon 

strated. Q.E.I. 

EXAMPLE 3. Let the line AGK be an hyperbola, having its asymptote 

NX perpendicular to the horizontal plane AK, to find the density of the 

medium that will make a projectile move in that line. 

Let MX be the other asymptote, meeting 
the ordinate DG produced in V ; and from 
the nature of the hyperbola, the rectangle of 
XV into VG will be given. There is also 
given the ratio of DN to VX, and therefore 
the rectangle of DN into VG is given. Let 
that be bb : and, completing the parallelo 
gram DNXZ, let BN be called a; BD, o; 
NX, c; and let the given ratio of VZ to 

ZX or DN be -. Then DN will be equal 




MA. BD K N 

bb 
to a o } VG equal to 



, VZ equal to X a o. and GD or NX 
a o n 



m 



m 



-VZ VG equal to c a + o . Let the term - be 

n n a o a o 

bb bb bb bb , 

resolved into the converging series ~^" + ^ + ^l 00 + ^4 > &c and 

m bb m bb bb bb 

GD will become equal to c - a - + -o ~ o ^ o 2 51 



SEC. II.] 



OF NATURAL PHILOSOPHY. 



273 



&c. The second term o o of this series is to be u?ed for Qo; the 

n aa 

third o 2 , with its sign changed for Ro 2 ; and the fourth o 3 , with its 

m bb bb bb 

sign changed also for So 3 , and their coefficients , and are to 

n aa a a 

be pat for Q, R, and S in the former rule. Which being done, the den- 

bb 
a* 



sity of the medium will come out as , , 



bb 



a 



mm 
nn 



2mbb 
naa 



I 



mm 



- , that is, if in VZ you take VY equal to 



aa 



aa 



1 m 2 

VG, as YT7- For aa and ^ a 2 



2mbb b 
nn n aa 

2mbb b 4 

H are the squares of XZ 

n aa 

and ZY. But the ratio of the resistance to gravity is found to be that of 
3XY to 2YG ; and the velocity is that with which the body would de- 

XY 2 

scribe a parabola, whose vertex is G, diameter DG, latus rectum ^ v . Sup 
pose, therefore, that the densities of the medium in each of the places G 
are reciprocally as the distances XY, and that the resistance in any place 
G is to the gravity as 3XY to 2YG ; and a body let go from the place A, 
with a due velocity, will describe that hyperbola AGK. Q.E.I. 

EXAMPLE 4. Suppose, mdeMtely, the line AGK to be an hyperbola 
described with the centre X, and the asymptotes MX, NX, so that, having 
constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G 
and its asymptote in V, VG may be reciprocally as any power DN n of the 
line ZX or DN, whose index is the number n : to find the density of the 
medium in which a projected body will describe this curve. 

For BN, BD, NX, put A, O, C, respec- ^ 

tively, and let VZ be to XZ or DN as d to 
bb 



e, and VG be equal to 



be equal to A O, VG == ^= 



then DN will 



VZ = 



O, and GD or NX VZ VG equal 




274 

term 



THE MATHEMATICAL PRINCIPLES [BOOK H 

nbb 



nn -f- n 



bb U ! J x * 

=rr be resolved into an infinite series -r- + 
A Of A" A. n 

3 -- 3nn + 2/i 



X O + 



n 



" ~ x bb O 3 ,&c,,andGD will be equal 



g^TT-T X 00 O 2 + 

c bb d nbb + ?m - 
toC -A--T-+-O- -r O - ~ 

e A" e A" + l 2A n -f 

+ H i^T t "\ bb 3 > &c - The second tcrm - O - -T 

6A n + e A n 4- l 

series is to be used for 0,0, the third ^ 66O 2 for Roo, the fourth 

-\~r~3 bbO 5 for So 3 . And thence the density of the medium 



Oof this 



-, in any place G 7 will be 



2dnbb 



nub* 



and therefore if in VZ you take VY equal to n X VG, that density is re- 

n w IT- j ^ * 2rf//66 /mfi 4 

ciprocally as XY. For A 2 and A 2 -- A + r are the 

tc/ o^x ./\_ " 

squares of XZ and ZY. Hut the resistance in the same place G is to the 
force of gravity as 3S X - to 4RR, that is, as XY to 



And the velocity there is the same wherewith the projected body would 
move in a parabola, whose vertex is G, diameter GD, and latus rectum 

2XY 2 

or - -------- - - . Q.E.I. 



R 



nn 



VG 




AC 
HT 



SCHOLIUM. 

In the same manner that the den 
sity of the medium comes out to be as 

S X AC . 

Tjr m ^ r - 1) if the resistance 

lx X HI 

is put as any power V" of the velocity 
V, the density of the medium will 



come out to be as 



B C D E Q 



. x 



S 



And therefore if a curve can be found, such that the ratio of to 

4 o 

R i- 



SEC. II.J 



OF NATURAL PHILOSOPHY, 



275 



n 1 



, or ofgr^ to 



may be given ; the body, in an uni- 



z 




HT 

AC 

form medium, whose resistance is as the power V" of the velocity V, will 
move in this curve. But let us return to more simple curves. 

Because there can be no motion in a para 
bola except in a non-resisting medium, but 
in the hyperbolas here described it is produced 
by a perpetual resistance ; it is evident that 
the line which a projectile describes in an 
uniformly resisting medium approaches nearer 
to these hyperbolas than to a parabola. That 
line is certainly of the hyperbolic kind, but 
about the vertex it is more distant from the 
asymptotes, and in the parts remote from the 

vertex draws nearer to them than these hy- M JL BD K 

perbolas here described. The difference, however, is not so great between 
the one and the other but that these latter may be commod^ously enough 
used in practice instead of the former. And perhaps these may prove more 
useful than an hyperbola that is more accurate, and at the same time more 
compounded. They may be made use of, then, in this manner. 

Complete the parallelogram XYGT, and the right line GT will touch 
the hyperbola in G, and therefore the density of the medium in G is re- 

GT 2 

ciprocally as the tangent GT, and the velocity there as ^ -^=- ; and the 




resistance is to the force of gravity as GT to 

Therefore if a body projected from the 
place A, in the direction of the right line 
AH, describes the hyperbola AGK and 
AH produced meets the asymptote NX in 
H, arid AI draw r ri parallel to it meets the 
other asymptote MX in I ; the density of 
the mediu.n in A will be reciprocally as 
AH. and the velocity of the body as -J 

AH 1 

. . . and the resis an^e there to the force 
Al 



2nn 



n +2 



^- X GV. 




. TT 2nn + 2n 

of gravity rs AH to ZiTo ~ 



X AI. Her,ce the following rules a e 



deduced. 

RULE 1. If the density of the medium at A, and the velocity with which 
the body is projected remain the same, and the angle NAH be changed , 
the lengths AH, AI, HX will remain. Therefore if those lengths, in any 



276 THE MATHEMATICAL PRINCIPLES [BOOK II. 

one case, are found, the hyperbola may afterwards be easily determined 
from any given angle NAH. 

RULE 2. If the angle NAH, and the density of the medium at A, re 
main the same, and the velocity with which the body is projected be 
changed, the length AH will continue the same ; and AI will be changed 
in a duplicate ratio of the velocity reciprocally. 

RULE 3. If the angle NAH, the velocity of the body at A, and the ac- 
celerative gravity remain the same, and the proportion of the resistance at 
A to the motive gravity be augmented in any ratio ; the proportion of AH 
to A I will be augmented in the same ratio, the latus rectum of the above- 

AH 2 

mentioned parabola remaining the same, and also the length propor- 

AI 

tional to it ; and therefore AH will be diminished in the same ratio, and 
AI will be diminished in the duplicate of that ratio. But the proportion 
of the resistance to the weight is augmented, when either the specific grav 
ity is made less, the magnitude remaining equal, or when the density of 
the medium is made greater, or when, by diminishing the magnitude, the 
resistance becomes diminished in a less ratio than the weight. 

RULE 4. Because the density of the medium is greater near the vertex 
of the hyperbola than it is in the place A, that a mean density may be 
preserved, the ratio of the least of the tangents GT to the tangent AH 
ought to be found, and the density in A augmented in a ratio a little 
greater than that of half the sum of those tangents to the least of the 
tangents GT. 

RULE 5. If the lengths AH, AI are given, and the figure AGK is to be 
described, produce HN to X, so that HX may be to AI as n -\- 1 to 1 ; and 
with the centre X, and the asymptotes MX, NX, describe an hyperbola 
through the point A, such that AI may be to any of the lines VG as XV" 

to xr. 

RULE 6. By how much the greater the number n is, so much the more 
accurate are these hyperbolas in the ascent of the body from A, and less 
accurate in its descent to K ; and the contrary. The conic hyperbola 
keeps a mean ratio between these, and is more simple than the rest. There 
fore if the hyperbola be of this kind, and you are to find the point K, 
where the projected body falls upon any right line AN passing through 
the point A, let AN produced meet the asymptotes MX, NX in M and N, 
and take NK equal to AM. 

RULE 7. And hence appears an expeditious method of determining this 
hyperbola from the phenomena. Let two similar and equal bodies be pro 
jected with the same velocity, in different angles HAK, h A k, and let them 
fall upon the plane of the horizon in K and k ; and note the proportion 
of AK to A A". Let it be as d to e. Then erecting a perpendicular A I of 
uny length, assume any how the length AH or Ah, and thence graphically, 



SEC. II. 



OF NATURAL PHILOSOPHY. 



27? 



or by scale and compass, collect the lengths AK, A/>* (by Rule 6). If the 
ratio of AK to A/.* bo the same with that of d to e, the length of AH was 





rightly assumed. If not, take on the indefinite right line SM, the length 
SM equal to the assumed AH ; and erect a perpendicular MN equal to the 

AK d 

difference -r-r of the ratios drawn into any given right line. By the 

like method, from several assumed lengths AH, you may find several points 
N ; and draw througli them all a regular curve NNXN, cutting tr.e right 
line SMMM in X. Lastly, assume AH equal to the abscissa SX, and 
thence find again the length AK ; and the lengths, w hich are to the as 
sumed length AI, and this last AH, as the length AK known by experi 
ment, to the length AK last found, will be the true lengths AI and AH, 
which were to be found. But these being given, there will be given also 
the resisting force of the medium in the place A, it being to the force of 
gravity as AH to JAI. Let the density of the medium be increased by 
Rule 4, and if the resisting force just found be increased in the same ratio, 
it will become still more accurate. 

RULE 8. The lengths AH, HX being found ; let there be now re 
quired the position of the line AH, according to which a projectile thrown 
with that given velocity shall fall upon any point K. At the joints A 
and K, erect the lines AC, KF perpendicular to the horizon : whereof let 
AC be drawn downwards, and be equal to AI or ^HX. With the asymp 
totes AK, KF, describe an hyperbola, whose conjugate shall pass through 
the point C ; and from the centre A, with the interval AH. describe a cir 
cle cutting that hyperbola in the point H ; then the projectile thrown in 
the direction of the right line AH will fall upon the point K. Q.E.I. For 
the point H, because of the given length AH, must be somewhere in the 
circumference of the described circle. Draw CH meeting AK and KF in 
E and F: and because CH, MX are parallel, and AC, AI equal, AE will 
be equal to AM, and therefore also equal to KN. But CE is to AE as 
FH to KN. and therefore CE and FH are equal. Therefore the point H 
falls upon the hyperbolic curve described with the asymptotes AK,.KF 
whose conjugate passes through the point C ; and is therefore found in the 



27S 



THE MATHEMATICAL PRINCIPLES 



[BOOK 11 




common intersection of this hyperbolic 
curve and the circumference of the de 
scribed circle. Q.E.D. It is to be ob 
served that this operation is the same, 
whether the right line AKN be parallel to 
the horizon, or inclined thereto in any an 
gle : and that from two intersections H, 
//., there arise two angles NAH, NAA ; 
and that in mechanical practice it is suf 
ficient once to describe a circle, then to 
apply a ruler CH, of an indeterminate length, HO to the point C, that its 
part PH, intercepted between the circle and the right line FK, may bo 
equal to its part CE placed between the point C and the right line AK 

What has been said of hyperbolas may be easily 
applied to pir i >;>l.i3. For if a parabola be re 
presented by XAGK, touched by a right line XV 
in the vertex X, and the ordinates IA, YG be as 
any powers XI", XV" ; of the abscissas XI, XV ; 
draw XT, GT, AH, whereof let XT be parallel 
to VG, and let GT, AH touch the parabola in 
G and A : and a body projected from any place 
A, in the direction of the right line AH, with a 
due velocity, will describe this parabola, if the density of the medium in 
each of the places G be reciprocally as the tangent GT. In that case the 
velocity in G will be the same as would cause a body, moving in a non- 
resisting space, to describe a conic parabola, having G for its vertex, VG 

2GT 2 

produced downwards for its diameter, and -. for its latus 

nn n X VG 

rectum. And the resisting force in G will be to the force of gravity as GT to 

2nti 2tt 

~2~ VG. Therefore if NAK represent an horizontal line, and botli 

the density of the medium at A, and the velocity with which the body is 
projected, remaining the same, the angle NAH be any how altered, the 
lengths AH, AI, HX will remain; and thence will be given the vertex X 
of the parabola, and the position of the right line XI ; and by taking VG 
to IA as XV n to XI", there will be given all the points G of the parabola, 
through which the projectile will pass. 




SEC. IILJ OF NATURAL PHILOSOPHY. 279 

SECTION III. 

Of the motions of bodies which are resisted partly in the ratio of the ve 
locities, and partly in the duplicate of the same ratio. 

PROPOSITION XI. THEOREM VIII. 

If a body be resisted partly in the ratio and partly in the duplicate ratio 
of its velocity, and moves in a similar medium by its innate force 
only; and the times be taken in arithmetical progression; then 
quantities reciprocally proportional to the velocities, increased by a cer 
tain given quantity, will be in geometrical progression. 
With the centre C, and the rectangular asymptotes ^ 

OADd and CH, describe an hyperbola BEe, and let | \p 

AB, DE, de. be parallel to the asymptote CH. In | 

the asymptote CD let A, G be given points ; and if 

the time be expounded by the hyperbolic area ABED 

uniformly increasing, I say, that the velocity may ~r 

be expressed by the length DF, whose reciprocal 

GD, together with the given line CG, compose the 

length CD increasing in a geometrical progression. 

For let the areola DEec/ be the least given increment of the time, and 

Dd will be reciprocally as DE, and therefore directly as CD. Therefore 

the decrement of ^TR, which (by Lem. II, Book II) is ^ no , will be also as 



D 
tf 



CD CG + GD 1 CG 

GO* r GD 2 ~ fc 1S>aS GD + GJD 2 * * nerefore tne timc 
uniformly increasing by the addition of the given particles EDcfe, it fol 

lows that r decreases in the same ratio with the velocity. For the de 



crement of the velocity is as the resistance, that is (by the supposition), as 
the sum of two quantities, whereof one is as the velocity, and the other as 

the square of the velocity ; and the decrement of ~~ is as the sum of the 



1 C^(^ 1 

quantities ~-^=r and pfp,> whereof the first is ^^r itself, and the last 



i i 

is a* /-TFT; therefore T^-R is as tne velocity, the decrements of both 

- CilJ 



being analogous. And if the quantity GD reciprocally proportional to 
T, be augmented by the given quantity CG ; the sum CD, the time 



ABED uniformly increasing, will increase !n a geometrical progression. 
Q.E.D. 



THE MATHEMATICAL PRINCIPLES [BOOK II 

COR. 1. Therefore, if, having the points A and G given, the time bo 
expounded by the hyperbolic area ABED, the velocity may be expounded 

by -r the reciprocal of GD. 






COR. 2. And by taking GA to GD as the reciprocal of the velocity at 
the beginning to the reciprocal of the velocity at the end of any time 
ABED, the point G will be found. And that point being found the ve 
locity may be found from any other time given. 

PROPOSITION XII. THEOREM IX. 

The same things being supposed, I say, that if the spaces described are 
taken in arithmetical progression, the velocities augmented by a cer 
tain given quantity will be in geometrical progression. 

In the asymptote CD let there be given the 
point R, and, erecting the perpendicular RS 
meeting the hyperbola in S, let the space de 
scribed be expounded by the hyperbolic area 
I RSED ; and the velocity will be as the length 

J GD, which, together with the given line CG, 

** composes a length CD decreasing in a geo 
metrical progression, while the space RSED increases in an arithmetical 
[(regression. 

For, because the incre nent EDde of the space is given, the lineola DC?, 
which is the decrement of GD, will be reciprocally as ED, and therefore 
directly as CD ; that is, as the sum of the same GD and the given length 
CG. But the decrement of the velocity, in a time reciprocally propor 
tional thereto, in which the given particle of space D^/eE is described, is 
as the resistance and the time conjunctly, that is. directly as the sum of 
two quantities, whereof one is as the velocity, the other as the square of 
the velocity, and inversely as the veh city ; and therefore directly as the 
sum of two quantities, one of which is given, the other is- as the velocity. 
Therefore the decrement both of the velocity and the line GD is as a given 
quantity and a decreasing quantity conjunctly; and, because the decre 
ments are analogous, the decreasing quantities will always be analogous ; 
viz., the velocity, and the line GD. U.E.D. 

COR. 1. If the velocity be expounded by the length GD, the space de 
scribed will be as the hyperbolic area DESR. 

COR. 2. And if the point be assumed any how, the point G will be 
found, by taking GR to GD as the velocity at the beginning to the velo 
city after any space RSED is described. The point G being given, the 
space is given from the given velocity : and the contrary. 

Cotw 3. Whence since (by Prop. XI) the velocity is given from the given 



SEC. Ilt.1 



Or NATURAL PHILOSOPHY. 



281 




time, and (by this Prop.) the space is given from the given velocity ; the 
space will be given from the given time : and the contrary. 

PROPOSITION XKI. THEOREM X. 

Supposing that a body attracted downwards by an uniform gravity as 
cends or descends in a right line; and that the same is resisted 
partly in the ratio of its velocity, and partly in the duplicate ratio 
thereof: I say, that, if right lines parallel to the diameters of a circle 
and an hyperbola, be drawn through the ends of the, conjugate diame 
ters, and the velocities be as some segments of those parallels drawn 
from a given point, the times will be as the sectors of the, areas cut 
off by right lines drawn from the centre to the ends of the segments ; 
and the contrary. 

CASE 1. Suppose first that the body is ascending, 
and from the centre D, with any semi-diameter DB, 
describe a quadrant BETF of a circle, and through 
the end B of the semi-diameter DB draw the indefi 
nite line BAP, parallel to the semi-diameter DF. In 
chat line let there be given the point A, and take the 
segment AP proportional to the velocity. And since 
one part of the resistance is as the velocity, and 
another part as the square of the velocity, let the 
whole resistance be as AP 2 -f 2BAP. Join DA, DP, cutting the circle 
in E and T, and let the gravity be expounded by DA 2 , so that the gravity 
shall be to the resistance in P as DA 2 to AP 2 +2BAP ; and the time of the 
whole ascent will be as the sector EDT of the circle. 

For draw DVQ,, cutting off the moment PQ, of the velocity AP, and the 
moment DTV of the sector DET answering to a given moment of time ; 
and that decrement PQ, of the velocity will be as the sum of the forces of 
gravity DA 2 and of resistance AP 2 + 2BAP, that is (by Prop. XII 
BookII,Elem.),asDP*. Then the arsa DPQ, which is proportional to PQ : 
is as DP 2 , and the area DTV, which is to the area DPQ, as DT 2 to DP 2 , it 
as the given quantity DT 2 . Therefore the area EDT decreases uniformly 
according to the rate of the future time, by subduction of given particles DT V, 
and is therefore proportional to the time of the whole ascent. Q..E.D. 

CASE 2. If the velocity in the ascent 
of the body be expounded by the length 
AP as before, and the resistance be made 
as AP 2 -f- 2BAP,and if the force of grav 
ity be less than can be expressed by DA 2 ; 
take BD of such a length, that AB 2 
BD 2 maybe proportional to the gravity, 
and let DF be perpendicular and equal 




F O 



S2 THE MATHEMATICAL PRINCIPLES [BOOK ll. 

to DB, and through the vertex F describe the hyperbola FTVE, whose con 
jugate semi -diameters are DB and DF ; and which cuts DA in E, and DP, 
DQ in T and V ; and the time of the whole ascent will be as the hyper 
bolic sector TDE. 

For the decrement PQ of the velocity, produced in a given particle of 
time, is as the sum of the resistance AP 2 -f 2BAP and of the gravity 
AB 2 BD 2 , that is, as BP 2 BD 2 . But the area DTV is to the area 
DPQ as DT 2 to DP 2 ; and, therefore, if GT be drawn perpendicular to 
DF. as GT 2 or GD 2 DF 2 to BD 2 , and as GD 2 to BP 2 , and, by di 
vision, as DF 2 to BP 2 BD 2 . Therefore since the area DPQ is as PQ, 
that is, as BP 2 BD 2 , the area DTV will be as the given quantity DF 2 . 
Therefore the area EDT decreases uniformly in each of the equal particles 
of time, by the subduction of so many given particles DTV, and therefore 
is proportional to the time. Q.E.D. 

CASE 3. Let AP be the velocity in the descent of 
""" the body, and AP 2 + 2BAP the force of resistance, 
and BD 2 AB 2 the force of gravity, the angle DBA 
being a right one. And if with the centre D, and the 
principal vertex B, there be described a rectangular 
hyperbola BETV cutting DA, DP, and DQ produced 
in E, T, and V : the sector DET of this hyperbola will 
D be as the whole time of descent. 

For the increment PQ of the velocity, and the area DPQ proportional 
to it, is as the excess of the gravity above the resistance, that is, as 
m)2 ? _ AB 2 _2BAP AP 2 or BD 2 BP 2 . And the area DTV 
is to the area DPQ as DT 3 to DP 2 ; and therefore as GT 2 or GD" - 
BD 2 to BP 2 , and as GD 2 to BD 2 , and, by division, as BD 2 to BD 2 - 
BP 2 . Therefore since the ami DPQ is as BD 2 BP 2 , the area DTV 
will be as the given quantity BD 2 . Therefore the area EDT increases 
uniformly in the several equal particles of time by the addition of as 
many given particles DTV, and therefore is proportional to the time of 
the descent. Q.E.D. 

Con. If with the centre D and the semi-diameter DA there be drawn 
through the vertex A an arc A/ similar to the arc ET, and similarly sub- 
tendino^the angle A DT, the velocity AP will be to the velocity which the 
body in the time EDT, in a non-resisting space, can lose in its ascent, or 
acquire in its descent, as the area of the triangle DAP to the area of the 
Bector DA/ ; and therefore is given from the time given. For the velocity 
ir a non-resistin^ medium is proportional to the time, and therefore to this 
sector : in a resisting medium, it is as the triangle ; and in both mediums, 
where it is least, it approaches to the ratio of equality, as the sector and 
triangle do 




SEC. III.] 



OF NATURAL PHILOSOPHY. 



283 



SCHOLIUM. 

One may demonstrate also that case in the ascent of the body, where the 
force of gravity is less than can be expressed by DA 2 or AB 2 + BD 2 , and 
greater than can be expressed by AB 2 DB 2 , and must be expressed by 
AB 2 . But I hasten to other things 

PROPOSITION XIV. THEOREM XL 

The same things being supposed, 1 say, that the space described in the 
ascent or descent is as the difference of the area by which the time is 
expressed, and of some other area which is augmented or diminished 
in an arithmetical progression ; if the forces compounded of the re 
sistance and the gravity be taken, in a geometrical progression. 
Take AC (in these three figures) proportional to the gravity, and AK 

to the resistance ; but take them on the same side of the point A, if the 

\* 

"1 



\ 



B A 



K QP 





body is descending, otherwise on the contrary. Erect A b, which make to 
DB as DB 2 to 4BAC : and to the rectangular asymptotes CK, CH, de 
scribe the hyperbola 6N ; and, erecting KN perpendicular to CK, the area 
A/AK will be augmented or diminished in an arithmetical progression, 
while the forces CK are taken in a geometrical progression. I say, there 
fore, that the distance of the body from its greatest altitude is as the excess 
of the area A6NK above the area DET. 

For since AK is as the resistance, that is, as AP 2 X 2BAP ; assume 



any given quantity Z, and put AK equal to 



then (by Lem, 



284 



THE MATHEMATICAL PRINCIPLES [BOOK II 

2APQ, + 2B A X PU 



II of this Book) the moment KL of AK will be equal to 
2BPQ 



or 



Z 



-, and the moment KLON of the area ANK will be equal to 



2BPQ X LO BPQ, X BD 3 
~~Z~~ >r 2Z X OK x~AB" 

CASE 1. Now if the body ascends, and the gravity be as AB 2 + BD 9 

BET being a circle, the line AC, which is proportional to the gravity 

AW2 i RF)2 

will be - ~ -- , and DP 2 or AP 2 + 2BAP + AB 2 + BD 2 will be 

AK XZ + AC X Z or CK X Z : and therefore the area DTV will be to 
the area DPQ, as DT 2 or I)B 2 to CK X Z. 

CASE 2. If the body ascends, and the gravity be as AB 2 BD 2 , the 
A r>2 _ HI) 2 

line AC will be " -- ^ --- , and DT 2 will be to DP 2 as DF 2 or DB 2 
Z 

to BP 2 BD 2 or AP 2 + 2BAP + AB 2 BD 2 , that is, to AK X Z + 



H 





AC X Z or CK X Z. And therefore the area DTV will be to the area 
DPQ as DB 2 to CK X Z. 

CASE 3. And by the same reasoning, if the body descends, and therefore 
the gravity is as BD 2 AB 2 , and the line AC becomes equal to 

or) 2 AB 2 

5T r ; the area DTV will be to the area DPQ as DB 2 to CK X 

Z 

Z : as above. 

Since, therefore, these areas are always in this ratio, if for the area 



SEC. 111^ OF NATURAL PHILOSOPHY. 285 

DTY, by which the moment of the time, always equal to itself, is express 
ed, there be put any determinate rectangle, as BD X m, the area DPQ,, 
that is, |BD X PQ, will be to BD X mas CK X Z to BD 2 . And thence 
PQ X BD 3 becomes equal to 2 BD XmX CK X Z, and the moment KLON 

BP X BD X tn 
of the area A6NK, found before, becomes - .-^ -- . From the area 



DET subduct its moment DTV or BD X m, and there will remain 
--- -Pp . Therefore the difference of the moments, that is, the 

AP X BD X m 

mo.nent of the difference of tne areas, is equal to -- 7-5 --- ; and 

therefore (because of the given quantity --- T-~ ) as the velocity AP ; 

that is, as the moment of the space which the body describes in its ascent 
or descent. And therefore the difference of the areas, and that space, in 
creasing or decreasing by proportional moments, and beginning together or 
vanishing together, are proportional. Q.E.D. 

COR. If the length, which arises by applying the area DET to the line 
BD, be called M ; and another length V be taken in that ratio to the length 
M, which the line DA has to the line DE; the space which a body, in a 
resisting medium, describes in its whole ascent or descent, will be to the 
space which a body, in a non-resisting medium, falling from rest, can de 
scribe in the same time, as the difference of the aforesaid areas to 

BD X V 2 
- -TO"""" j an( ^ therefore is given from the time given. For the space in a 

A.LJ 

non-resisting medium is in a duplicate ratio of the time, or as V 2 ; and. 

BD X V 2 
because BD and AB are given, as --- -TTT- . This area is equal to the 

DA 2 X BD x M 2 
area -- fvG r *~~~T~R "~ anc * ^ ne momen t Of M is m ; and therefore the 

DA 2 X BD X 2M X m 

moment of this area is --- =- --- ^5 -. But this moment is to 

"" X .A D 



the moment of the difference of the aforesaid areas DET and A6NK, viz., to 

AP X BD X m DA 2 X BD X M x DA 2 . ^^ 

- -- , as - -r- - to |BD X AP, or as into DET 



to DAP ; and, therefore, when the areas DKT and DAP are least, in the 

BD X V 2 
ratio of equality. Therefore the area r^ -- and the difference of the 

areas DET and A&NK, when all these areas are least, have equal moments ; 
and { re therefore equal. Therefore since the velocities, and therefore also 
the s] aces in both mediums described together, in the beginning of the de 
scent or the end of the ascent, approach to equality, and therefore are then 



286 THE MATHEMATICAL PRINCIPLES [BOOK II 

BD X V 2 

one to another as the area r-^ , and the difference of the areas DET 

AD 

and A6NK ; and moreover since the space, in a non-resisting medium, is 

BD X V 2 

perpetually as Tu~~> an( ^ tne s P ace > i n a resisting medium, is perpetu 
ally as the difference of the areas DET and A&NK ; it necessarily follows, 
that the spaces, in both mediums, described in any equal times, are one to 

BD X V 2 
another as that area 7-5 an( ^ ^he difference of the areas DET and 

A6NK. Q.E.D. 

SCHOLIUM. 

The resistance of spherical bodies in fluids arises partly from the tena 
city, partly from the attrition, and partly from the density of the medium. 
And that part of the resistance which arises from the density of the fluid 
is, as I said, in a duplicate ratio of the velocity ; the other part, which 
arises from the tenacity of the fluid, is uniform, or as the moment of the 
time ; and, therefore, we might now proceed to the motion of bodies, whicli 
are resisted partly by an uniform force, or in the ratio of the moments of 
the time, and partly in the duplicate ratio of the velocity. But it is suf 
ficient to have cleared the way to this speculation in Prop. VIII and IX 
foregoing, and their Corollaries. For in those Propositions, instead of the 
uniform resistance made to an ascending body arising from its gravity, 
one may substitute the uniform resistance which arises from the tenacity 
of the medium, when the body moves by its vis insita alone ; and when the 
body ascends in a right line, add this uniform resistance to the force of 
gravity, and subduct it when the body descends in a right line. One 
might also go on to the motion of bodies which are resisted in part uni 
formly, in part in the ratio of the velocity, and in part in the duplicate 
ratio of the same velocity. And I have opened a way to this in Prop. 
XIII and XIV foregoing, in which the uniform resistance arising from the 
tenacity of the medium may be substituted for the force of gravity, or be 
compounded with it as before. But I hasten to other things. 




SKC. -IV .] OF NATUEAL PHILOSOPHY. 2S? 

SECTION IV. 

Of the circular motion of bodies in resisting mediums. 

LEMMA III. 

Let PQR be a spiral rutting all the radii SP, SO, SR, <J*c., in equal 
angles. Draw tfie right line PT touching the spiral in any point P, 
and cutting the radius SQ in T ; cfo er?0 PO, QO perpendicular to 
the spiral, and meeting- in, O, and join SO. .1 say, that if Hie points 
P a/*(/ Q approach and coincide, the angle PSO vri/Z become a right 
angle, and the ultimate ratio of the rectangle TQ, X 2PS to P^ 3 //>i// 
/>e /ie ya/io o/" equality. 
For from the right angles OPQ, OQR, sub 
duct the equal angles SPQ, SQR, and there 
will remain the equal angles OPS, OQS. 
Therefore a circle which passes through the 
points OSP will pass also through the point 
Q. Let the points P and Q, coincide, and 
this circle will touch the spiral in the place 
of coincidence PQ, and will therefore cut the 
right line OP perpendicularly. Therefore OP will become a diameter of 
this circle, and the angle OSP, being in a semi-circle, becomes a right 
one. Q.E.1). 

Draw Q,D, SE perpendicular to OP, and the ultimate ratios of the lines 
will be as follows : TO to PD as TS or PS to PE, or 2PO to 2PS and 
PD to PO as PO to 2PO ; and, ex cequo pertorbatt, to TO to PO as PO 
to 2PS. Whence PO 2 becomes equal to TO X 2PS. O.E.D. 

PROPOSITION XV. THEOREM XII. 

Tf the density of a medium in each place thereof be recipr on iJl y as the 
distance of the places from an immovable centre, aud the centripetal 
force be in the duplicate ratio of the density ; I say, that a body mny 
revolve in a spiral which cuts all the radii drawn from that centre 
in a given angle. 

Suppose every thing to be as in the forego 
ing Lemma, and produce SO to V so that SV 
may be equal to SP. In any time let a body, 
in a resisting medium, describe the least arc 
PO, and in double the time the least arc PR : 
and the decrements of those arcs arising from 
the resistance, or their differences from the 
arcs which would be described in a non-resist 
ing medium in the same times, will be to each 
other as the squares of the times in which they 
are generated ; therefore the decrement of the 




288 THE MATHEMATICAL PRINCIPLES [_BoOK 11 

arc PQ is the fourth part of the decrement of the arc PR. Whence also 
if the area QSr be taken equal to the area PSQ, the decrement of the arc 
PQ will be equal to half the lineola Rr ; and therefore the force of resist 
ance and the centripetal force are to each other as the lineola jRrandTQ 
which they generate in the same time. Because the centripetal force with 
which the body is urged in P is reciprocally as SP 2 , and (by Lem. X, 
Book I) the lineola TQ, which is generated by that force, is in a ratio 
compounded of the ratio of this force and the duplicate ratio of the time 
in which the arc PQ, is described (for in this case I neglect the resistance, 
as being infinitely less than the centripetal force), it follows that TQ X 
SP 2 , that is (by the last Lemma), fPQ 2 X SP, will be in a duplicate ra 
tio of the time, and therefore the time is as PQ, X v/SP ; and the velo 
city of the body, with which the arc PQ is described in that time, as 

PQ 1 

-p or , that is, in the subduplicate ratio of SP reciprocally. 



And, by a like reasoning, the velocity with which the arc QR is described, 
is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ and 
QR are as the describing velocities to each other ; that is, in the subdu 



plicate ratio of SQ to SP, or as SQ to x/SP X SQ; and, because of the 
equal angles SPQ, SQ? , and the equal areas PSQ, QSr, the arc PQ is to 
the arc Qr as SQ to SP. Take the differences of the proportional conse 
quents, and the arc PQ will be to the arc Rr as SQ to SP VSP X ~SQ~, 
or ^VQ. For the points P and Q coinciding, the ultimate ratio of SP 



X SQ to |VQ is the ratio of equality. Because the decrement of 
the arc PQ arising from the resistance, or its double Rr, is as the resistance 

and the square of the time conjunctly, the resistance will be &Sp-^r op. 

* 1 



X 
But PQ was to Rr as SQ to |VQ, and thence SSaTXToD becomes as 

Jr vst X oJr 

-VQ -OS 

pWxsvxSQ, or ns ETp^TsP- For the poillts p and a coincidin & 

SP and SQ coincide also, and the angle PVQ becomes a right one; and, 
because of the similar triangles PVQ, PSO, PQ. becomes to -VQ as OP 

OS 

to | OS. Therefore : y -- is as the resistance, that is, in the ratio of 
\J i X ol 

the density of the medium in P and the duplicate ratio of the velocity 
conjunc-tly. Subduct the duplicate ratio of the velocity, namely, the ratio 

1 OS 

^5, and there will remain the density of the medium in P. as 7^5 - = 

OA Ur X fei 

Let the spiral be given, and ; because of the given ratio of OS to OP, the 
density of the medium in P will be as ~-p. Therefore in a medium whose 



SEC. IV,] OF NATURAL PHILOSOPHY. 2S9 

density is reciprocally as SP the distance from the centre, a body will re 
volve in this spiral. Q.E.D. 

COR. 1. The velocity in any place P, is always the same wherewith a 
body in a non-resisting medium with the same centripetal force would re 
volve in a circle, at the same distance SP from the centre. 

COR. 2. The density of the medium, if the distance SP be given, is as 

OS OS 

TTp, but if that distance is not given, as ^ ^5. And thence a spiral 

may be fitted to any density of the medium. 

COR. 3. The force of the resistance in any place P is to the centripetal 
force in the same place as |OS to OP. For those forces are to each other 

^VQ x PQ iPQ 2 

as iRr and TQ, or as 1 ^-^~- and ^-, that is, as iVQ and PQ, 

ol%, ol 

or |OS and OP. The spiral therefore being given, there is given the pro 
portion of the resistance to the centripetal force ; and, vice versa, from that 
proportion given the spiral is given. 

COR. 4. Therefore the body cannot revolve in this spiral, except where 
the force of resistance is less than half the centripetal force. Let the re 
sistance be made equal to half the centripetal force, and the spiral will co 
incide with the right line PS, and in that right line the body will descend 
to the centre with a velocity that is to the velocity, with which it was 
proved before, in the case of the parabola (Theor. X, Book I), the descent 
would be made in a non-resisting medium, in the subduplicate ratio of 
unity to the number two. And the times of the descent will be here recip 
rocally as the velocities, and therefore given. 

COR. 5. And because at equal distances 
from the centre the velocity is the same in the 
spiral PQ,R as it is in the right line SP, and 
the length of the spiral is to the length of the 
right line PS in a given ratio, namely, in the 
ratio of OP to OS ; the time of the descent in 
the spiral will be to the time of the descent in 
the right line SP in the same given ratio, and 
therefore given. 

COR. 6. If from the centre S, with any two 
given intervals, two circles are described ; and 
these circles remaining, the angle which the spiral makes with the radius" 
PS be any how changed ; the number of revolutions which the body can 
complete in the space between the circumferences of those circles, going 

PS 

round in the spiral from one circumference to another, will be as 7^, or as 

Ok5 

ths tangent of the angle which the spiral makes with the radius PS ; and 

19 





290 THE MATHEMATICAL PRINCIPLES [BOOK II 

OP 

the time of the same revolutions will be as -^, that is, as the secant of the 

Uo 

same angle, or reciprocally as the density of the medium. 

COR. 7. If a body, in a medium whose density is reciprocally as the dis 
tances of places from the centre, revolves in any curve AEB about that 
centre, and cuts the first radius AS in the same 
angle in B as it did before in A, and that with a 
velocity that shall be to its first velocity in A re 
ciprocally in a subduplicate ratio of the distances 
from the centre (that is, as AS to a mean propor 
tional between AS and BS) that body will con 
tinue to describe innumerable similar revolution? 
BFC, CGD, &c., and by its intersections will 
distinguish the radius AS into parts AS, BS, CS, DS, &c., that are con 
tinually proportional. But the times of the revolutions will be as the 
perimeters of the orbits AEB, BFC, CGD, &c., directly, and the velocities 

3 3 

at the beginnings A, B, C of those orbits inversely ; that is as AS % BS % 

CS" 2 ". And the whole time in which the body will arrive at the centre, 
will be to the time of the first revolution as the sum of all the continued 

142 

proportionals AS 2 , BS 2 , CS 2 , going on ad itifinitum, to the first term 

* i 3 

AS 2 ; that is, as the first term AS 2 to the difference of the two first AS 2 

BS% or as f AS to AB very nearly. Whence the whole time may be 
easily found. 

COR. 8. From hence also may be deduced, near enough, the motions of 
bodies in mediums whose density is either uniform, or observes any other 
assigned law. From the centre S, with intervals SA, SB, SC, &c., con 
tinually proportional, describe as many circles ; and suppose the time of 
the revolutions between the perimeters of any two of those circles, in the 
medium whereof we treated, to be to the time of the revolutions between 
the same in the medium proposed as the mean density of the proposed me 
dium between those circles to the mean density of the medium whereof we 
treated, between the same circles, nearly : and that the secant of the angle 
in which the spiral above determined, in the medium whereof we treated, 
cuts the radius AS, is in the same ratio to the secant of the angle in which 
the new spiral, in the proposed medium, cuts the same radius : and also 
that the number of all the revolutions between the same two circles is nearly 
as the tangents of those angles. If this be done every where between every 
two circles, the motion will be continued through all the circles. And by 
this means one may without difficulty conceive at what rate and in what 
time bodies ought to revolve in any regular medium. 



SEC. IY.1 OF NATURAL PHILOSOPHY. 291 

COR. 9. And although these motions becoming eccentrical should be 
performed in spirals approaching to an oval figure, yet, conceiving the 
several revolutions of those spirals to be at the same distances from each 
other, and to approach to the centre by the same degrees as the spiral above 
described, we may also understand how the motions of bodies may be per 
formed in spirals of that kind. 

PROPOSITION XVI. THEOREM XIII. 

If the density of the medium in each of the places be reciprocally as the 
distance of the >, places from the immoveable centre, and the centripetal 
force be reciprocally as any power of the same distance, I say, that the 
body may revolve in a spiral intersecting all the radii drawn from 
that centre in a given, angle. 
This is demonstrated in the same manner as 

the foregoing Proposition. For if the centri 

petal force in P be reciprocally as any power 

SP n + 1 of the distance SP whose index is n 

+ 1 ; it will be collected, as above, that the 

time in which the body describes any arc PQ, 

i 
will be as PQ, X PS 2U ; and the resistance in 



i!! x _ 

n; raS "~ 



X SP n; PQ, X SP"XSQ, 




, , 1 in X OS . 1 \n X OS . 

therefore as Qp"^~gpirqTT tliat 1S > ( because - ~~Qp~~ 1S a lven 

quantity), reciprocally as SP n + ! . And therefore, since the velocity is recip 
rocally as SP 3 ", the density in P will be reciprocally as SP. 

COR. 1. The resistance is to the centripetal force as 1 ^//. X OS 
to OP. 

COR. 2. If the centripetal force be reciprocally as SP 3 . 1 w will be 
=== ; and therefore the resistance and density of the medium will be 
nothing, as in Prop. IX, Book I. 

COR. 3. If the centripetal force be reciprocally as any power of the ra 
dius SP, whose index is greater than the number 3, the affirmative resist 
ance will be changed into a negative. 

SCHOLIUM. 

This Proposition and the former, which relate to mediums of unequal 
density, are to be understood of the motion of bodies that are so small, that 
the greater density of the medium on one side of the body above that on 
the other is not to be considered. I suppose also the resistance, cateris 
paribus, to be proportional to its density. Whence, in mediums whose 



292 THE MATHEMATICAL PRINCIPLES | BoOK II 

force of resistance is not as the density, the density must be so much aug 
mented or diminished, that either the excess of the resistance may be taken 
away, or the defect supplied. 

PROPOSITION XVII. PROBLEM IV 

To find the centripetal force and the resisting force of the medium, by 
which a body, the law of the velocity being given, shall revolve in a 
given spiral. 

Let that spiral be PQR. From the velocity, 
with which the body goes over the very small arc 
PQ,, the time will be given : and from the altitude 
TQ,, which is as the centripetal force, and the 
square of the time, that force will be given. Then 
from the difference RSr of the areas PSQ, and 
Q,SR described in equal particles of time, the re 
tardation of the body will be given ; and from 
the retardation will be found the resisting force 
and density of the medium. 

PROPOSITION XVIII. PROBLEM V. 

The law of centripptal force being given, to find the density of the me 
dium in each of the places thereof, by which a body may describe a 
given spiral. 

From the centripetal force the velocity in each place must be found ; 
then from the retardation of the velocity the density of the medium is 
found, as in the foregoing Proposition. 

But I have explained the method of managing these Problems in the 
tenth Proposition and second Lemma of this Book; and will no longer 
detain the reader in these perplexed disquisitions. I shall now add some 
things relating to the forces of progressive bodies, and to the density and 
resistance of those mediums in which the motions hitherto treated of, and 
those akin to them, are performed. 




SEC. V.] OF NATURAL PHILOSOPHY. 293 

SECTION V. 

Of the density and compression of fluids ; and of hydrostatics. 

THE DEFINITION OF A FLUID. 

A fluid is any body whose parts yield to any force impressed on it, 
by yielding, are easily moved among themselves. 

PROPOSITION XIX. THEOREM XIv 

All the parts of a homogeneous and unmoved fluid included in any un 
moved vessel, and compressed on every side (setting aside the consider 
ation of condensation, gravity, and all centripetal forces], will be 
equally pressed on every side, and remain in their places without any 
motion arising from that pressure. 
CASE 1. Let a fluid be included in the spherical 
vessel ABC, and uniformly compressed on every 
side : 1 say, that no part of it will be moved by 
that pressure. For if any part, as L), be moved, 
all such parts at the same distance from the centre 
on every side must necessarily be moved at the 
same time by a like motion ; because the pressure 
of them all is similar and equal ; and all other 
motion is excluded that does not arise from that 
pressure. But if these parts come all of them nearer to the centre, the 
fluid must be condensed towards the centre, contrary to the supposition. 
If they recede from it, the fluid must be condensed towards the circumfer 
ence ; which is also contrary to the supposition. Neither can they move 
in any one direction retaining their distance from the centre, because for 
the same reason, they may move in a contrary direction : but the sami 
part cannot be moved contrary ways at the same time. Therefore no 
part of the fluid will be moved from its place. Q,.E.T). 

CASE 2. I say now, that all the spherical parts of this fluid are equally 
pressed on every side. For let EF be a spherical part of the fluid ; if this 
be not pressed equally on every side, augment the lesser pressure till it be 
pressed equally on every side; and its parts (by Case I) will remain in 
their places. But before the increase of the pressure, they would remain 
in their places (by Case 1) ; and by the addition of a new pressure they 
will be moved, by the definition of a fluid, from those places. Now these 
two conclusions contradict each other. Therefore it was false to say that 
the sphere EF was not pressed equally on every side. Q,.E.D. 

CASE 3. I say besides, that different spherical parts have equal pressures. 
For the contiguous spherical parts press each other mutually and equally 
in the point of contact (by Law III). But (by Case 2) they are pressed on 
every side with the same force. Therefore any two spherical parts lot 




391 THE MATHEMATICAL PRINCIPLES [BoOK II. 

contiguous, since an intermediate spherical part can touch both, will be 
pressed with the same force. Q.E.D. 

CASE 4. I say now, that all the parts of the fluid are every where press 
ed equally. For any two parts may be touched by spherical parts in any 
points whatever ; and there they will equally .press those spherical parts 
(by Case 3). and are reciprocally equally pressed by them (by Law III). 
Q.E.D. 

CASE 5. Since, therefore, any part GHI of the fluid is inclosed by the 
rest of the fluid as in a vessel, and is equally pressed on every side ; and 
also its parts equally press one another, and are at rest among themselves ; 
it is manifest that all the parts of any fluid as GHI, which is pressed 
equally on every side, do press each other mutually and equally, and are at 
rest among themselves. Q.E.D. 

CASE 6. Therefore if that fluid be included in a vessel of a yielding 
substance, or that is not rigid, and be not equally pressed on every side, 
the same will give way to a stronger pressure, by the Definition of fluidity. 

CASE 7. And therefore, in an inflexible or rigid vessel, a fluid will not 
Sustain a stronger pressure on one side than on the other, but will give 
way to it, and that in a moment of time ; because the rigid side of the 
vessel does not follow the yielding liquor. But the fluid, by thus yielding, 
will press against the opposite side, and so the pressure will tend on every 
side to equality. And because the fluid, as soon as it endeavours to recede 
from the part that is most pressed, is withstood by the resistance of the 
vessel on the opposite side, the pressure will on every side be reduced to 
equality, in a moment of time, without any local motion : and from thence 
the parts of the fluid (by Case 5) will press each other mutually and equal 
ly, and be at rest among themselves. Q..E.D. 

COR. Whence neither will a motion of the parts of the fluid among 
themselves be changed by a pressure communicated to the external super 
ficies, except so far as either the figure of the superficies may be somewhere 
altered, or that all the parts of the fluid, by pressing one another more in 
tensely or remissly, may slide with more or less difficulty among them 
selves. 

PROPOSITION XX. THEOREM XV. 

Jf all the parts of a spherical fluid, homogeneous at equal distances from 
the centre, lying on. a spherical concentric bottom, gravitate towards 
the centre of the whole, the bottom will sustain the weight of a cylin 
der, whose base is equal to the superficies of the bottom, and whose al 
titude is the same with that of the incumbent fluid. 
Let OHM be the superficies of the bottom, and AEI the upper super 
ficies of the fluid. Let the fluid be distinguished into concentric orbs of 
3qual thickness, by the innumerable spherical superficies *3PK, CGL : and 




SEC. V OF NATURAL PHILOSOPHY. 295 

conceive the force of gravity to act only in the 
upper superficies of every orb, and the actions 
to be equal on the equal parts of all the su 
perficies. Therefore the upper superficies AE 
is pressed by the single force of its own grav 
ity, by which all the parts of the upper orb, 
and the second superficies BFK, will (by 
Prop. XIX), according to its measure, be 
equally pressed. The second superficies BFK 
is pressed likewise by the force of its own 
gravity, which, added to the former force, 
makes the pressure double. The third superficies CGL is, according to its 
measure, acted on by this pressure and the force of its own gravity besides, 
which makes its pressure triple. And in like manner the fourth superfi 
cies receives a quadruple pressure, the fifth superficies a quintuple, and so 
on. Therefore the pressure acting on every superficies is not as the solid 
quantity of the incumbent fluid, but as the number of the orbs reaching 
to the upper surface of the fluid ; and is equal to the gravity of the lowest 
orb multiplied by the number of orbs : that is, to the gravity of a solid 
whose ultimate ratio to the cylinder above-mentioned (when the number of 
the orbs is increased and their thickness diminished, ad infiititum, so that 
the action of gravity from the lowest superficies to the uppermost may be- 
some continued) is the ratio of equality. Therefore the lowest superficies 
sustains the weight of the cylinder above determined. Q..E.D. And by a 
like reasoning the Proposition will be evident, where the gravity of the 
fluid decreases in any assigned ratio of the distance from the centre, and 
also where the fluid is more rare above and denser below. Q.E.D. 

COR. 1. Therefore the bottom is not pressed by the whole weight of the 
incumbent fluid, but only sustains that part of it which is described in the 
Proposition ; the rest of the weight being sustained archwise by the spheri 
cal figure of the fluid. 

COR. 2. The quantity of the pressure is the same always at equal dis 
tances from the centre, whether the superficies pressed be parallel to the 
horizon, or perpendicular, or oblique ; or whether the fluid, continued up 
wards from the compressed superficies, rises perpendicularly in a rectilinear 
direction, or creeps obliquely through crooked cavities and canals, whether 
those passages be regular or irregular, wide or narrow. That the pressure 
is not altered by any of these circumstances, may be collected by applying 
the demonstration of this Theorem to the several cases of fluids. 

COR. 3. From the same demonstration it may also be collected (by Prop. 
XIX), that the parts of a heavy fluid acquire no motion among themselvei 
by the pressure of the incumbent veight, except that motion which arises 
from condensation. 



296 THE MATHEMATICAL PRINCIPLES [BCOK II 

Con. 4. And therefore if another body of the same specific gravity, in 
capable of condensation, be immersed in this fluid, it will acquire no mo 
tion by the pressure of the incumbent weight: it will neither descend nor . 
ascend, nor change its figure. If it be spherical, it will remain so, notwith 
standing the pressure ; if it be square, it will remain square; and that, 
whether it be soft or fluid : whether it swims freely in the fluid, or lies at 
the bottom. For any internal part of a fluid is in the same state with the 
submersed body ; and the case of all submersed bodies that have the same 
magnitude, figure, and specific gravity, is alike. If a submersed body, re 
taining its weight, should dissolve and put on the form of a fluid, this 
body, if before it would have ascended, descended, or from any pressure as 
sume a new figure, would now likewise ascend, descend, or put on a new 
figure ; and that, because its gravity and the other causes of its motion 
remain. But (by Case 5, Prop. XIX; it would now be at rest, and retain 
its figure. Therefore also in the former case. 

COR. 5. Therefore a body that is specifically heavier than a fluid con 
tiguous to it will sink ; and that which is specifically lighter will ascend, 
and attain so much motion and change of figure as that excess or defect of 
gravity is able to produce. For that excess or defect is the same thing as an 
impulse, by which a body, otherwise in equilibria with the parts of the 
fluid, is acted on: and may be compared with the excess or defect of a 
weight in one of the scales of a balance. 

COR. 6. Therefore bodies placed in fluids have a twofold gravity the 
one true and absolute, the other apparent, vulgar, and comparative. Ab 
solute gravity is the whole force with which the body tends downwards ; 
relative and vulgar gravity is the excess of gravity with which the body 
tends downwards more than the ambient fluid. By the first kind of grav 
ity the parts of all fluids and bodies gravitate in their proper places ; and 
therefore their weights taken together compose the weight of the whole. 
For the whole taken together is heavy, as may be experienced in vessels 
full of liquor ; and the weight of the whole is equal to the weights of all 
the parts, and is therefore composed of them. By the other kind of grav 
ity bodies do not gravitate in their places ; that is, compared with one 
another, they do not preponderate, but, hindering one another s endeavours 
to descend, remain in their proper places, as if they were not heavy. Those 
things which are in the air, and do not preponderate, are commonly looked 
on as not heavy. Those which do preponderate are commonly reckoned 
heavy, in as much as they are not sustained by the weight of the air. The 
Common weights are nothing else but the excess of the true weights above 
the weight of the air. Hence also, vulgarly, those things are called light 
which are less heavy, and, by yielding to the preponderating air, mount 
upwards. But these are only comparatively lig s &mA not truly so, because 

hey descend in racuo. Thus, in water, bodies *>icfc. by their greater or 



SEC. V.] OF NATURAL PHILOSOPHY. 297 

less gravity, descend or ascend, are comparatively and apparently heavy or 
light ; and their comparative and apparent gravity or levity is the excess 
.or defect by which their true gravity either exceeds the gravity of the 
water or is exceeded by it. But those things which neither by preponder 
ating descend, nor, by yielding to the preponderating fluid, ascend, although 
by their true weight they do increase the weight of the whole, yet com 
paratively, and in the sense of the vulgar, they do not gravitate in the wa 
ter. For these cases are alike demonstrated. 

COR. 7. These things which have been demonstrated concerning gravity 
take place in any other centripetal forces. 

COR. 8. Therefore if the medium in which any body moves be acted on 
either by its own gravity, or by any other centripetal force, and the body 
be urged more powerfully by the same force ; the difference of the forces is 
that very motive force, which, in the foregoing Propositions, I have con 
sidered as a centripetal force. But if the body be more lightly urged by 
that force, the difference of the forces becomes a centrifugal force, and is tc 
be considered as such. 

COR. 9. But since fluids by pressing the included bodies do not 
change their external figures, it appears also (by Cor. Prop. XIX) that they 
will not change the situation of their internal parts in relation to onf 
another ; and therefore if animals were immersed therein, and that all sen 
sation did arise from the motion of their parts, the fluid will neither hurt 
the immersed bodies, nor excite any sensation, unless so far as those bodies 
may be condensed by the compression. And the case is the same of any 
system of bodies encompassed with a compressing fluid. All the parts of 
the system will be agitated with the same motions as if they were placed 
in a vacuum, and would only retain their comparative gravity ; unless so 
far as the fluid may somewhat resist their motions, or be requisite to con- 
glutinate them by compression. 

PROPOSITION XXI. THEOREM XVI. 

<et the density of any fluid be proportional to the compression, and its 
parts be attracted downwards by a centripetal force reciprocally pro 
portional to the distances from the centre : I say, that, if those dis 
tances be taken continually proportional, the densities of the fluid at 
the same distances will be also continually proportional. 
Let ATV denote the spherical bottom of the fluid, S the centre, S A, SB. 
SC, SD, SE, SF, &c., distances continually proportional. Erect the per 
pendiculars AH, BI, CK, DL, EM, PN, &c., which shall be as the densi 
ties of the medium in the places A, B, C, D, E, F : and the specific grav 

ATT RT f^K" 
ities in those places will be aa -r-, , - &c., or, which is all one, a&- 



298 



THE MATHEMATICAL PRINCIPLES 



[BOOK II. 




AH BI CK 

ATT BC CD Suppose, first, these gravities to be uniformly continued 

from A to B, from B to C, from C to D, (fee., the decrements in the points 
B, C, D, (fee., being taken by steps. Arid these gravi 
ties drawn into the altitudes AB, BC, CD, (fee., will 
give the pressures AH, BI, CK, (fee., by which the bot 
tom ATV is acted on (by Theor. XV). Therefore the 
particle A sustains all the pressures AH, BI, CK, DJL, 
(fee., proceeding in infinitum ; and the particle B sus 
tains the pressures of all but the first AH ; and the par 
ticle C all but the two first AH, BI ; and so on : and 
therefore the density AH of the first particle A is to 
the density BI of the second particle B as the sum of 
all AH -f- BI + CK + DL, in infinitum, to the sum of 
all BI -f- CK -f DL, (fee. And BI the density of the second particle B is 
to CK the density of the third C, as the sum of all BI -f CK + DL, (fee., 
to the sum of all CK -f- DL, (fee. Therefore these sums are proportional 
to their differences AH, BI, CK, (fee., and therefore continually propor 
tional (by Lem. 1 of this Book) ; and therefore the differences AH, BI, 
CK, (fee., proportional to the sums, are also continually proportional. 
Wherefore since the densities in the places A, B, C, (fee., are as AH, BI, 
CK, (fee., they will also be continually proportional. Proceed intermis- 
sively, and, ex ccquo, at the distances SA, SC, SE, continially proportional, 
the densities AH, CK, EM will be continually proportional. And by the 
same reasoning, at any distances SA, SD, SG, continually proportional, 
the densities AH. DL, GO, will be continually proportional. Let now the 
points A, B, C. D, E, (fee., coincide, so that the progression of the specif .c 
gravities from the bottom A to the top of the fluid may be made continual ; 
and at any distances SA, SD, SG, continually proportional, the densities 
AH, DL, GO, being all along continually proportional, will still remain 
continually proportional. Q.E.D. 

COR. Hence if the density of the fluid in two places, 
as A and E, be given, its density in any other place Q, 
may be collected. With the centre S, and the rectan 
gular asymptotes SQ, SX, describe an hyperbola cut 
ting the perpendiculars AH, EM, QT in , e, and q } 
as also the perpendiculars HX, MY, TZ, let fall upon 
the asypmtote SX, in //, m, and t. Make the area 
Y////Z to the given area YmAX as the given area 
EeqQ to the given area EeaA ; and the line Z produced will cut off the 
line Q,T. proportional to the density. For if the lines SA, SE, SQ are 
continually proportional, the areas ReqQ., fyaA will be equal, and thence 




X 



SEC. V. 



OF NATURAL PHILOSOPHY. 



299 



the areas YwYZ. X/zwY, proportional to them, will be also equal ; and 
the lines SX, SY, SZ, that is, AH, EM, Q,T continually proportional, as 
they ought to be. And if the lines SA, SE, SQ, 5 obtain any other order 
in the series of continued proportionals, the lines AH, EM, Q,T, because 
of the proportional hyperbolic areas, will obtain the same order in another 
series of quantities continually proportional. 



PROPOSITION XXII THEOREM XVII. 

Let the density of any fluid be proportional to the compression, and its 
parts be attracted downwards by a gravitation reciprocally propor 
tional to the squares of the distances from the centre : I say, that if 
the distances be taken in harmonic progression, the densities of the 
fluid at those distances will be in a geometrical progression. 
Let S denote the centre, and SA, 

SB, SC, SD, SE, the distances in 

geometrical progression. Erect the 

perpendiculars AH, BI, CK, (fee., 

which shall be as the densities of c 

the fluid in the places A, B, C, D, 

E, (fee., and the specific gravities 

thereof in those places will be as 



AH BI 



,^-, (fee. Suppose these 



V 


LN 


M 


I. 


K 


i 


V 




V 






V 








V7 










^ 


n 


w 
r 


/ 





L / 


/ 


/t 





SA 2 SB 2 SC 2 

gravities to be uniformly continued, the first from A to B, the second from 
B to C, the third from C to I), &c. And these drawn into the altitudes 
AB, BC, CO, DE, (fec. ; or, which is the same thing/into the distances SA, 

ATT r>T OT7" 

SB, SC, (fee., proportional to those altitudes, will give -~-r-, ^=5, -~~, (fee.. 

the exponents of the pressures. Therefore since the densities are as th^ 
sums of those pressures, the differences AH BI, BI CK, (fee., of tb, 

densities will be as the differences of those sums ~-r~, ^, ~~, (fee. With 

the centre S, and the asymptotes SA, S#, describe any hyperbola, cutting 
the perpendiculars AH, BI, CK, (fee., in a, 6, c, (fee., and the perpendicu 
lars H/, I//,, K?#, let fall upon the asymptote Sv, in h, i, k ; and the dif 
ferences of the densities tu, uw, (fee., will be as A , ^^, (fee. And the 



SA ; SB ; 



rectangles tu X th, uw X uij (fee., or tp, uq, (fee., as 
that is, as Aa, Bb, (fee. 



AH X th BI X ui 

, (fee. 



SA SB 

For, by the nature of the hyperbola, SA is to AH 

or St as th to Ar, and therefore pri is equal to Aa . And, by a like 



SA 



300 THE MATHEMATICAL PRINC. PLES [BOOK II. 

reasoning, ^n~~ * s e( l ua ^ to ^, & c - But Aa > B ^> ^c, & c v are continu 

ally proportional, and therefore proportional to their differences Aa B&, 
B6 Cc ; &c., therefore the rectangles fy?, nq, &c., are proportional to those 
differences ; as also the sums of the rectangles tp + uq, or tp + uq -f w 
to the sums of the differences Aa Cc or Aa Da 7 . Suppose several of 
these terms, and the sum of all the differences, as Aa F/, will be pro 
portional t? the sum of all the rectangles, as zthn. Increase the number 
of terms, and diminish the distances of the points A, B, C, (fee., in iiijini- 
tum, and those rectangles will become equal to the hyperbolic area zthn. 
and therefore the difference Aa F/ 19 proportional to this area. Take 
now T any distances, as SA, SD, SF, in harmonic progression, and the dif 
ferences Aa Da 7 , Da 1 F/ will be equal ; and therefore the areas thlx, 
xlnz, proportional to those differences will be equal among themselves, and 
the densities St, S:r, Sz, that is, AH, DL, FN, continually proportional. 
Q.E.D. 

COR. Hence if any two densities of the fluid, as AH and BI, be given, 
the area thiu, answering to their difference tu, will be given; and thence 
the density FN will be found at any height SF, by taking the area thnz to 
that given area thiu as the difference Aa F/ to the difference Aa Eh. 

SCHOLIUM. 

By a like reasoning it may be proved, that if the gravity of the particles 
of a fluid be diminished in a triplicate ratio of the distances from the centre ; 
and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely, 

SA 3 SA 3 SA 3 . 

opt ^ e ta ^ en m an arithmetical progression, the densities AH. 



BI, CK, &c., will be in a geometrical progression. And if the gravity be 
diminished in a quadruplicate ratio of the distances, and the reciprocals of 

the cubes of the distances (as ^-r^, SRS sps ^ c ^ ^ e ta ^ cn ^ n ai> i tnmet i- 

cai progression, the densities AH, BI, CK, &c., will be in geometrical pro 
gression. And so in irtfinitum. Again ; if the gravity of the particles of 
the fluid be the same at all distances, and the distances be in arithmetical 
progression, the densities will be in a geometrical progression as Dr. Hal- 
ley has found. If the gravity be as the distance, and the squares of the 
distances be in arithmetical progression, the densities will be in geometri 
cal progression. And so in infinitum. These things will be so, when the 
density of the fluid condensed by compression is as the force of compres 
sion ; or, which is the same thing, when the space possessed by the fluid is 
reciprocally as this force. Other laws of condensation may be supposed, 
as that the cube of the compressing force may be as the biquadrate of the 



SEC. V.] OF NATURAL PHILOSOPHY. 301 

de isity ; or the triplicate ratio of tlie force the same with the quadruplicate 
ratio of the density : in which case, if the gravity he reciprocally as the 
square of the distance from the centre, the density will be reciprocally at 
the cube of the distance. Suppose that the cube of the compressing force 
be as the quadrato-cube of the density ; and if the gravity be reciprocally 
as the square of the distance, the density will be reciprocally in a sesqui- 
plicate ratio of the distance. Suppose the compressing force to be in a du 
plicate ratio of the density, and the gravity reciprocally in a duplicate ra 
tio of the distance, and the density will be reciprocally as the distance. 
To run over all the cases that might bo offered would be tedious. But as 
to our own air, this is certain from experiment, that its density is either 
accurately, or very nearly at least, as the compressing force ; and therefore 
the density of the air in the atmosphere of the earth is as the weight of 
the whole incumbent air, that is, as the height of the mercury in the ba 
rometer. 

PROPOSITION XXIII. THEOREM XVIII. 

If a fluid be composed of particles mutually flying each other, and the 
drnsity be as the compression, the centrifugal forces of the particles 
will be reciprocally proportional to tlie distances of their centres. And, 
vice versa, particles flying each otli,er, with forces that are reciprocally 
proportional to the distances of their centres^ compose an elastic fluid, 
whose density is as the compression. 
Let the fluid be supposed to be included in a cubic 
space ACE, and then to be reduced by compression into 
a lesser cubic space ace ; and the distances of the par- F 
tides retaining a like situation with respect to each 
other in both the spaces, will be as the sides AB, ab of 
the cubes ; and the densities of the mediums will be re 
ciprocally as the containing spaces AB 3 , ab 3 . In the 
plane side of the greater cube A BCD take the square 
DP equal to the plane side db of the lesser cube: and, 
by the supposition, the pressure with which the square 
DP urges the inclosed fluid will be to the pressure with 
which that square db urges the inclosed fluid as the densities of the me 
diums are to each other, that is, asa/> 3 to AB 3 . But the pressure with 
which the square DB urges the included fluid is to the pressure with which 
the square DP urges the same fluid as the square DB to the square DP, 
that is, as AB 2 to ab z . Therefore, ex cequo, the pressure with which the 
square DB urges the fluid is to the pressure with which the square db 
urges the fluid as ab to AB. Let the planes FGH,/V?, U drawn through 
the middles of the two cubes, and divide the fluid into tw^/ parts, These 
parts will press each other mutually with the same forces with which they 



A 




THE MATHEMATICAL PRINCIPLES [BOOK II. 

are themselves pressed by the planes AC, ac, that is, in the proportion of 
ab to AB : arid therefore the centrifugal forces by which these pressures 
are sustained are in the same ratio. The number of the particles being 
equal, and the situation alike, in both cubes, the forces which all the par 
ticles exert, according to the planes FGH,/o7/,, upon all, are as the forces 
which each exerts on each. Therefore the forces which each exerts on 
each, according to the plane FGH in the greater cube, are to the forces 
which each exerts on each, according to the plane fgh in the lesser cube, 
us ab to AB,*that is, reciprocally as the distances of the particles from each 
other. Q.E.D. 

And, vice versa, if the forces of the single particles are reciprocally as 
the distances, that is, reciprocally as the sides of the cubes AB, ab ; the 
sums of the forces will be in the same ratio, and the pressures of the sides 
i)B, db as the sums of the forces ; and the pressure of the square DP to 
the pressure of the side DB as ab 2 to AB 2 . And, ex cequo, the pressure of 
the square DP to the pressure of the side db as ab* to AB 3 ; that is, the 
force of compression in the one to the force of compression in the other as 
the density in the former to the density in the latter. Q.E.D. 

SCHOLIUM. 

By a like reasoning, if the centrifugal forces of the particles are recip 
rocally in the duplicate ratio of the distances between the centres, the cubes 
of the compressing forces will be as the biquadrates of the densities. If 
the centrifugal forces be reciprocally in the triplicate or quadruplicate ratio 
of the distances, the cubes of the compressing forces will be as the quadrato- 
cubes, or cubo-cubes of the densities. And universally, if D be put for the 
distance, and E for the density of the compressed fluid, and the centrifugal 
forces be reciprocally as any power D n of the distance, whose index is the 
number ??, the compressing forces will be as the cube roots of the power 
E n + 2 . whose index is the number n + 2 ; and the contrary. All these 
things are to be understood of particles whose centrifugal forces terminate 
in those particles that are next them, or are diffused not much further. 
We have an example of this in magnetical bodies. Their attractive vir 
tue is terminated nearly in bodies of their own kind that are next them. 
The virtue of the magnet is contracted by the interposition of an iron 
plate, and is almost terminated at it : for bodies further off are not attracted 
by the magnet so much as by the iron plate. If in this manner particles repel 
others of their own kind that lie next them, but do not exert their virtue 
on the more remote, particles of this kind will compose such fluids as are 
treated of in this Proposition, If the virtue of any particle diffuse itself 
every way in inftnitum, there will be required a greater force to produce 
an equal condensation of a greater quantity of the flui 1. But whether 



SEC. VI.] OF NATURAL PHILOSOPHY. 303 

elastic fluids do really consist of particles so repelling each other, is a phy 
sical question. We have here demonstrated mathematically the property 
of fluids consisting of particles of this kind, that hence philosophers may 
take occasion to discuss that question. 



SECTION VI. 

Of the motion and resistance of funependulous bodies. 

PROPOSITION XXIV. THEOREM XIX. 

The quantities of matter i/i funependulous bodies, whose centres of oscil 
lation are equally distant from, the centre of suspension, are in a, ratio 
compounded of the ratio of the weights and the duplicate ratio of the 
times of the oscillations in vacuo. 

For the velocity which a given force can generate in a given matter in 
a given time is as the force and the time directly, and the matter inversely. 
The greater the force or the time is, or the less the matter, the greater ve 
locity will he generated. This is manifest from the second Law of Mo 
tion. Now if pendulums are of the same length, the motive forces in places 
equally distant from the perpendicular are as the weights : and therefore 
if two bodies by oscillating describe equal arcs, and those arcs are divided 
into equal parts ; since the times in which the bodies describe each of the 
correspondent parts of the arcs are as the times of the whole oscillations, 
the velocities in the correspondent parts of the oscillations will be to each 
other as the motive forces and the whole times of the oscillations directly, 
and the quantities of matter reciprocally : and therefore the quantities of 
matter are as the forces and the times of the oscillations directly and the 
velocities reciprocally. But the velocities reciprocally are as the times, 
and therefore the times directly and the velocities reciprocally are as the 
squares of the times; and therefore the quantities of matter are as the mo 
tive forces and the squares of the times, that is, as the weights and the 
squares of the times. Q.E.D. 

COR. 1. Therefore if the times are equal, the quantities of matter in 
each of the bodies are as the weights. 

COR. 2. If the weights are equal, the quantities of matter will be as the 
pquarcs of the times. 

COR. 3. If the quantities of matter are equal, the weights will be recip 
rocally as the squares of the times. 

COR. 4. Whence since the squares of the times, cceteris paribus, are as 
the length* of the pendulums, therefore if both the times and quantities of 
matter are equal, the weights will be as the lengths of the pendulums. 



J04 THE MATHEMATICAL PRINCIPLES [BOOK 11 

COR. 5. And universally, the quantity of matter in the pendulous body 
is as the weight and the square of the time directly, and the length of the 
pendulum inversely. 

COR. 6. But in a non-resisting medium, the quantity of matter in the 
pendulous body is as the comparative weight and the square of the time 
directly, and the length of the pendulum inversely. For the comparative 
weight is the motive force of the body in any heavy medium, as was shewn 
above ; and therefore does the same thing in such a non-resisting medium 
as the absolute weight does in a vacuum. 

COR. 7. And hence appears a method both of comparing bodies one 
among another, as to the quantity of matter in each ; and of comparing 
the weights of the same body in different places, to know the variation of 
its gravity. And by experiments made with the greatest accuracy, I 
have always found the quantity of matter in bodies to be proportional to 
their weight. 

PROPOSITION XXV. THEOREM XX. 

Funependulous bodies that are, in, any medium, resisted in the ratio oj 
the moments of time, and funepetidulons bodies that move in a non- 
resisting medium of the same specific gravity, perform their oscilla 
tions in. a cycloid in the same time, and describe proportional parts oj 
arcs together. 

Let AB be an arc of a cycloid, which 
a body D, by vibrating in a non-re 
sisting medium, shall describe in any 
time. Bisect that arc in C, so that C 
may be the lowest point thereof ; and 
the accelerative force with which the 
body is urged in any place D, or d or 
E, will be as the length of the arc CD, 




pressed by that same arc ; and since the resistance is as the moment of the 
time, and therefore given, let it ba expressed by the given part CO of the 
cycloidal arc, and take the arc Od in the same ratio to the arc CD that 
the arc OB has to the arc CB : and the force with which the body in d is 
urged in a resisting medium, being the excess of the force Cd above the 
resistance CO, will be expressed by the arc Od, and will therefore be to 
the force with which the body D is urged in a non-resisting medium in the 
place D, as the arc Od to the arc CD ; and therefore also in the place B, 
as the arc OB to the arc CB. Therefore if two bodies D, d go from the place 
B, and are urged by these forces ; since the forces at the beginning are as 
the arc CB and OB, the first velocities and arcs first described will be in 
the same ratio. Let those arcs be BD and Ed, and the remaining arcf 



SEC. VI. | OF NATURAL PHILOSOPHY. 305 

CD, Odj will be in the same ratio. Therefore the forces, being propor 
tional to those arcs CD, Od, will remain in the same ratio as at the be 
ginning, and therefore the bodies will continue describing together arcs in 
the same ratio. Therefore the forces and velocities and the remaining arcs 
CD. Od, will be always as the whole arcs CB, OB, and therefore those re 
maining arcs wLl be described together. Therefore the two bodies D and 
d will arrive together at the places C and O ; that whicli moves in the 
non-resisting medium, at the place C, and the other, in the resisting me 
dium, at the place O. Now since the velocities in C and O are as the arcs 
CB, OB, the arcs which the bodies describe when they go farther will be 
in the same ratio. Let those arcs be CE and Oe. The force with which 
the body D in a non-resisting medium is retarded in E is as CE, and the 
force with which the body d in the resisting medium is retarded in e, is as 
the sum of the force Ce and the resistance CO, that is, as Oe ; and there 
fore the forces with which the bodies are retarded are as the arcs CB, OB, 
proportional to the arcs CE, Oe ; and therefore the velocities, retarded in 
that given ratio, remain in the same given ratio. Therefore the velocities 
and the arcs described with those velocities are always to each other in 
that given ratio of the arcs CB and OB ; and therefore if the entire arcs 
AB, aB are taken in the same ratio, the bodies D andc/ will describe those 
aics together, and in the places A and a will lose all their motion together. 
Therefore the whole oscillations are isochronal, or are performed in equal 
times ; and any parts of the arcs, as BD, Ed, or BE, Be, that are described 
together, are proportional to the whole arcs BA, B. Q,.E.D. 

COR. Therefore the swiftest motion in a resisting medium does not fall 
upon the lowest point C, but is found in that point O, in which the whole 
arc described Ba is bisected. And the body, proceeding from thence to a, 
is retarded at the same rate with which it was accelerated before in its de 
scent from B to O. 

PROPOSITION XXVI. THEOREM XXI. 

Funependulous bodies, that are resisted in the ratio of the velocity, have 

their oscillations in a cycloid isochronal. 

For if two bodies, equally distant from their centres of suspension, de 
scribe, in oscillating, unequal arcs, and the velocities in the correspondent 
parts of the arcs be to each other as the whole arcs ; the resistances, pro 
portional to the velocities, will be also to each other as the same arcs. 
Therefore if these resistances be subducted from or added to the motive 
forces arising from gravity which are as the same arcs, the differences or 
sums will be to each other in the same ratio of the arcs ; and since the in 
crements and decrements of the velocities are as these differences or sums, 
the velocities will be always as the whole arcs; therefore if the velocities 
are in any one case as the whole arcs, they will remain always in the same 

20 



306 THE MATHEMATICAL PRINCIPLES [BOOK. 11 

ratio. But at the beginning of the motion, when the bodies begin to de 
scend and describe those arcs, the forces, which at that time are proportional 
to the arcs, will generate velocities proportional to the arcs. Therefore 
the velocities will be always as the whole arcs to be described, and there 
fore those arcs will be described in the same time. Q,.E.D. 



PROPOSITION XXVII. THEOREM XXII. 

If fnnependulous bodies are resisted in the duplicate ratio of their 
velocities, the differences between the times of the oscillations in a re 
sisting medium, and the times of the oscillations in a non-resisting 
medium of the same specific gravity, will be proportional to the arcs 
described in oscillating nearly. 

For let equal pendulums in a re 
sisting medium describe the unequal 
arcs A, B ; and the resistance of the 
body in the arc A will be to the resist 
ance of the body in the correspondent 
part of the arc B in the duplicate ra 
tio of the velocities, that is, as, A A to 
BB nearly. If the resistance in the 
arc B were to the resistance in the arc 
A as AB to AA, the times in the arcs A and B would be equal (by the last 
Prop.) Therefore the resistance AA in the arc A, or AB in the arc B, 
causes the excess of the time in the arc A above the time in a non-resisting 
medium ; and the resistance BB causes the excess of the time in the arc B 
above the time in a non-resisting medium. But those excesses are as the 
efficient forces AB and BB nearly, that is, as the arcs A and B. Q.E.D. 

COR, 1. Hence from the times of the oscillations in unequal arcs in a 
resisting medium, may be know r n the times of the oscillations in a non- re 
sisting medium of the same specific gravity. For the difference of the 
times will be to the excess of the time in the lesser arc above the time in a 
non-resisting medium as the difference of the arcs to the lesser arc. 

COR. 2. The shorter oscillations are more isochronal, and very short 
ones are performed nearly in the same times as in a non-resisting medium. 
But the times of those which are performed in greater arcs are a little 
greater, because the resistance in the descent of the body, by which the 
time is prolonged, is greater, in proportion to the length described in the 
descent than the resistance in the subsequent ascent, by which the time is 
contracted. But the time of the oscillations, both short arid long, seems to 
be prolonged in some measure by the motion of the medium. For retard 
ed bodies are resisted somewhat less in proportion to the velocity, and ac 
celerated bodies somewhat more than those that proceed uniformly forwards ; 




SEC. VI.] 



OF NATURAL PHILOSOPHY. 



307 



because the medium, by the motion it has received from the bodies, going 
forwards the same way with them, is more agitated in the former case, and 
less in the latter ; and so conspires more or less with the bodies moved. 
Therefore it resists the pendulums in their descent more, and in their as 
cent less, than in proportion to the velocity; and these two causes concur 
ring prolong the time. 

PROPOSITION XXVIII. THEOREM XXIII. 

If afunependvlous body, oscillating in a cycloid, be resisted in the rati > 
of the moments of the time, its resistance will be to the force of grav 
ity as the excess of the arc described in the whole descent above the 
arc described in the subsequent ascent to twice the length of the pen 
dulum. 

Let BC represent the arc described 
in the descent, Ca the arc described in 
the ascent, and Aa the difference of 
the arcs : and things remaining as they 
were constructed and demonstrated in 
Prop. XXV, the force with which the 
oscillating body is urged in any place 
D will be to the force of resistance as 
the arc CD to the arc CO, which is 
half of that difference Aa. Therefore the force with which the oscillating 
body is urged at the beginning or the highest point of the cycloid, that is, 
the force of gravity, will be to the resistance as the arc of the cycloid, be 
tween that highest point and lowest point C, is to the arc CO ; that is 
(doubling those arcs), as the whole cycloidal arc, or twice the length of the 
pendulum, to the arc Aa. Q.E.D. 

PROPOSITION XXIX. PROBLEM VI. 

Supposing that a body oscillating in a cycloid is resisted in a duplicate 

ratio of the velocity: to find the resistance in each place. 
Let Ba be an arc described in one entire oscillation, C the lowest point 




C O 




K 



O ,S P rR Q M 

of the cycloid, and CZ half the whole cycloidal arc, equal to the length of 
the pendulum ; and let it be required to find the resistance of the body is 



30S THE MATHEMATICAL PRINCIPLES [BOOK 1L 

any place D. Cut the indefinite right line OQ in the points O, S, P, Q,, 
so that (erecting the perpendiculars OK, ST, PI, QE, and with the centre 
O, and the aysmptotcs OK, OQ, describing the hyperbola TIGE cutting 
the perpendiculars ST, PI, QE in T. I, and E, and through the point I 
drawing KF. parallel to the asymptote OQ, meeting the asymptote OK i i 
K, and the perpendiculars ST and QE in L and F) the hyperbolic area 
PIEQ may be to the hyperbolic area PITS as the arc BC, described in the 
descent of the body, to the arc Ca described in the ascent ; and that the 
area IEF may be to the area ILT as OQ to OS. Then with the perpen 
dicular MN cut off the hyperbolic area PINM, and let that area be to the 
hyperbolic area PIEQ as the arc CZ to the arc BC described in the de 
scent. And if the perpendicular RG cut off the hyperbolic area PIGR, 
which shall be to the area PIEQ as any arc CD to the arc BC described 
in the whole descent, the resistance in any place D will be to the force of 
OR 

gravity as the area IEF IGH to the area PINM. 



For since the forces arising from gravity with which the body is 
urged in the places Z, B, D, a, are as the arcs CZ. CB, CD, Ca and those 
arcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be the 
exponents both of the arcs and of the forces respectively. Let DC? be a 
very small space described by the body in its descent : and let it be expressed 
r >y the very small area RGor comprehended between the parallels RG, rg ; 
and produce r<? to //, so that GYlhg- and RGr may be the contemporane 
ous decrements of the areas IGH, PIGR. And the increment Gllhg 

IEF, or Rr X HG -^ IEF, of the area ~ IEF IGH will be 



, 
OQ OQ 

IFF 

to the decrement RGr, or Rr X RG, of the area PIGR, as HG - - 



OR 

to RG ; and therefore as OR X HG IEF to OR X GR or OP X 



PL that is (because of the equal quantities OR X HG, OR X HR OR 
X GR, ORHK OPIK, PIHR and PIGR + IGH), as PIGR + IGH 
OR OR 

IEF to OPIK. Therefore if the area - IEF IGH be called 



OQ 

Y, and RGgr the decrement of the area PIGR be given, the increment of 

the area Y will be as PIGR Y. 

Then if V represent the force arising from the gravity, proportional to 
the arc CD to be described, by which the body is acted upon in D, and R 
be put for the resistance, V R will be the whole force with which the 
body is urged in D. Therefore the increment of the velocity is as V R 
and the particle of time in which it is generated conjunctly. But the ve 
locity itself is as the contempo] aueous increment of the space described di- 



SEC. VI.J 



OF NATURAL PHILOSOPHY. 



309 



rectly and the same particle of time inversely. Therefore, since the re 
sistance is, by the supposition, as the square of the velocity, the increment 
of the resistance will (by Lem. II) be as the velocity and the increment of 
the velocity conjunctly, that is, as the moment of the space and V R 
conjunctly ; and, therefore, if the moment of the space be given, as V 
11 ; that is, if for the force V we put its exponent PIGR, and the resist 
ance R be expressed by any other area Z ; as PIGR Z. v 

Therefore the area PIGR uniformly decreasing by the subduction of 
given moments, the area Y increases in proportion of PIGR Y, and 
the area Z in proportion of PIGR Z. And therefore if the areas 
Y and Z begin together, and at the beginning are equal, these, by the 
addition of equal moments, will continue to be equal and in like man 
ner decreasing by equal moments, \vill vanish together. And, vice versa, 
if they together begin and vanish, they will have equal moments and te 
always equal ; and that, because if the resistance Z be augmented, the ve 
locity together with the arc C, described in the ascent of the body, will be 
diminished ; and the point in which all the motion together with the re 
sistance ceases coming nearer to the point C, the resistance vanishes sooner 
than the area Y. And the contrary will happen when the resistance is 
diminished. 

Now the area Z begins and ends where the resistance is nothing, that is, 
at the beginning of the motion where the arc CD is equal to the arc CB, 




K 



/IK 



O S P /~R Q M 

and the right line RG falls upon the right line Q.E ; and at the end of 
the motion where the arc CD is equal to the arc Ca, and RG falls upon 

the right line ST. And the area* Y or IEF IGH begins and ends 

also where the resistance is nothing, and therefore where IEF and 

IGH are equal ; that is (by the construction), where the right line RG 
falls successively upon the right lines Q,E and ST. Therefore those areas 
begin and vanish together, and are therefore always equal. Therefore the area 
OR 

IEF IGH is equal to the area Z, by which the resistance is ex 



pressed, and therefore is to the area PINM, by which the gravity is ex 
pressed, as the resistance to the gravity. Q.E.D. 



310 



THE MATHEMATICAL PRINCIPLES 



[BOOK 11. 



COR. 1 . Therefore the resistance in the lowest place C is to the force 

OP 

of gravity as the area ^ ~ IEF to the area PINM. 

COR. 2. But it becomes greatest where the area PIHR is to the area 
IEF as OR to OQ. For in that case its moment (that is, PIGR Y) 
becomes nothing. 

COR. 3. Hence also may be known the velocity in each place, as being 
in the subduplicate ratio of the resistance, and at the beginning of the mo 
tion equal to the velocity of the body oscillating in the same cycloid with 
out any resistance. 

However, by reason of the difficulty of the calculation by which the re 
sistance and the velocity are found by this Proposition, we have thought 
fit to subjoin the Proposition following. 

PROPOSITION XXX. THEOREM XXIV. 

If a right line aB be equal to the arc of a cycloid which an oscillating 
body describes, and at each of its points D the perpendiculars DK be 
erected, which shall be to the length of the pendulum as the resistance 
of the body in the corresponding points of the arc to the force of grav 
ity ; I say, that the difference between the arc described in the whole 
descent and the arc described in the whole subsequent ascent drawn 
into half the sum of the same arcs will be equal to the area BKa 
which all those perpendiculars take up. 

Let the arc of the cycloid, de 
scribed in one entire oscillation, be 
expressed by the right line aB, 
equal to it, and the arc which 
would have been described in vaciw 
by the length AB. Bisect AB in 
C, and the point C will represent 
the lowest point of the cycloid, and 
CD Mill be as the force arising from gravity, with which the body in D i,s 
urged in the direction of the tangent of the cycloid, and will have the same 
ratio to the length of the pendulum as the force in D has to the force of 
gravity. Let that force, therefore, be expressed by that length CD, and 
the force of gravity by the length of the pendulum ; and if in DE you 
take DK in the same ratio to the length of the pendulum as the resistance 
has to the gravity, DK will be the exponent of the resistance. From the 
centre C with the interval CA or CB describe a semi-circle BEeA. Let 
the body describe, in the least time, the space Dd ; and, erecting the per 
pendiculars DE, de, meeting the circumference in E and e, they will be as 
the velocities which the body descending in vacuo from the point B would 
acquire in the places D and d. This appears by Prop, LII, Book L Let 




SEC. VLJ OF NATURAL PHILOSOPHY. 311 

therefore, these velocities be expressed by those perpendiculars DE, de ; 
arid let DF be the velocity which it acquires in D by falling from B in 
the resisting medium. And if from the centre C with the interval OF we 
describe the circle F/M meeting the right lines de and AB in / and M, 
then M will be the place to which it would thenceforward, without farther 
resistance, ascend, and (//"the velocity it would acquire in d. Whence, 
also, if FO- represent the moment of the velocity which the body D, in de 
scribing the least space DC/, loses by the resistance of the medium ; and 
CN be taken equal to Cg ; then will N be the place to which the body, if 
it met no farther resistance, would thenceforward ascend, and MN will be 
the decrement of the ascent arising from the loss of that velocity. Draw 
F/n perpendicular to df t and the decrement Fg of the velocity DF gener 
ated by the resistance DK will be to the increment fm of the same velo 
city, generated by the force CD, as the generating force DK to the gener 
ating force CD. But because of the similar triangles F////, Fhg, FDC, 
fm is to Fm or Dd as CD to DF ; and, ex ceqno, Fg to Dd as DK to 
DF. Also Fh is to Fg as DF to CF ; and, ex ax/uo perturbate, Fh or 
MN to Do 1 as DK to CF or CM ; and therefore the sum of all the MN X 
CM will be equal to the sum of all the Dd X DK. At the moveable 
point M suppose always a rectangular ordinate erected equal to the inde 
terminate CM, which by a continual motion is drawn into the whole 
length Aa ; and the trapezium described by that motion, or its equal, the 
rectangle Aa X |aB, will be equal to the sum of all the MN X CM, and 
therefore to the sum of all the Dd X DK, that is, to the area BKVTa 

O.E.D. 

COR. Hence from the law of resistance, and the difference Aa of the 
arcs Ca } CB, may be collected the proportion of the resistance to the grav 
ity nearly. 

For if the resistance DK be uniform, the figure BKTa will be a rec 
tangle under Ba and DK; and thence the rectangle under ^Ba and Aa 
will be equal to the rectangle under Ba and DK, and DK will be equal to 
jAa. Wherefore since DK is the exponent of the resistance, and the 
length of the pendulum the exponent of the gravity, the resistance will be 
to the gravity as \Aa to the length of the pendulum ; altogether as in 
Prop. XXVIII is demonstrated. 

If the resistance be as the velocity, the figure BKTa will be nearly an 
ellipsis. For if a body, in a non-resisting medium, by one entire oscilla 
tion, should describe the length BA, the velocity in any place D would be 
as the ordinate DE of the circle described on the diameter AB. There 
fore since Ea in the resisting medium, and BA in the non-resisting one, 
are described nearly in the same times ; and therefore the velocities in each 
of the points of Ba are to the velocities in the correspondent points of the 
length BA. nearly as Ba is to BA , the velocity in the point D in the re- 



312 



THE MATHEMATICAL PRINCIPLES 



[BJOK 11. 



sisting medium will be as the ordinate of the circle or ellipsis described 
upon the diameter Ba ; and therefore the figure BKVTa will be nearly ac 
ellipsis. Since the resistance is supposed proportional to the velocity, le\ 
OV be the exponent of the resistance in the middle point O ; and an ellip 
sis BRVSa described with the centre O, and the semi-axes OB, OV, will 
be nearly equal to the figure BKVTa, and to its equal the rectangle Act 
X BO. Therefore Aa X BO is to OV X BO as the area of this ellipsis 
to OV X BO; that is, Aa is to OV as the area of the semi-circle to the 
square of the radius, or as 1 1 to 7 nearly ; and, therefore, T 7 T Aa is to the 
length of the pendulum as the resistance of the oscillating body in O to 
its gravity. 

Now if the resistance DK be in the duplicate ratio of the velocity, the 
figure BKVTa will be almost a parabola having V for its vertex arid OV 
for its axis, and therefore will be nearly equal to the rectangle under f Ba 
and OV. Therefore the rectangle under |Ba and Aa is equal to the rec 
tangle f Ba X OV, and therefore OV is equal to f Aa ; and therefore the 
resistance in O made to the oscillating body is to its gravity as f Aa to the 
length of the pendulum. 

And I take these conclusions to be accurate enough for practical uses. 
For since an ellipsis or parabola BRVSa falls in with the figure BKVTa 
in the middle point V, that figure, if greater towards the part BRV or 
VSa than the other, is less towards the contrary part, and is therefore 
nearly equal to it. 

PROPOSITION XXXI. THEOREM XXV. 

If the resistance made to an oscillating body in each of the proportional 
parts of the arcs described be augmented or diminished in, a given ra 
tio, the difference between the arc described in the descent and the arc 
described in the subsequent ascent ivill be augmented or diminished in 
the same ratio. 

For that difference arises from 
the retardation of the pendulum 
by the resistance of the medium, 
and therefore is as the whole re 
tardation and the retarding resist 
ance proportional thereto. In the 
foregoing Proposition the rectan- 

M isr u c o .-/ n P gle under the right line ^aB and 

the difference Aa of the arcs CB, Ca, was equal to the area BKTa, And 
that area, if the length aB remains, is augmented or diminished in the ra 
tio of the ordinates DK ; that is, in the ratio of the resistance and is there 
fore as the length aB and the resistance conjunctly. And therefore the 
rectangle under Aa and |aB is as aB and the resistance conjunctly, anc 
therefore Aa is as the resistance. QJE.D. 




SEC. VI.l OF NATURAL PHILOSOPHY. 313 

COR. 1. Hence if the resistance be as the velocity, the difference of 
the arts in the same medium will be as the whole arc described : and the 
contrary. 

COR. 2. If the resistance be in the duplicate ratio of the velocity, that 
difference will be in the duplicate ratio of the whole arc : and the contrary. 

COR. 3. And universally, if the resistance be in the triplicate or any 
other ratio of the velocity, the difference will be in the same ratio of the. 
whole arc : and the contrary. 

COR. 4. If the resistance be partly in the simple ratio of the velocity, 
and partly in the duplicate ratio of the same, the difference will be partly 
in the ratio of the whole arc, and partly in the duplicate ratio of it: and 
the contrary. So that the law arid ratio of the resistance will be the 
same for the velocity as the law and ratio of that difference for the length 
of the arc. 

COR. 5. And therefore if a pendulum describe successively unequal arcs, 
and we can find the ratio of the increment or decrement of this difference 
for the length of the arc described, there will be had also the ratio of the 
increment or decrement of the resistance for a greater or less velocity. 

GENERAL SCHOLIUM. 

From these propositions we may find the resistance of mediums by pen 
dulums oscillating therein. I found the resistance of the air by the fol 
lowing experiments. I suspended a wooden globe or ball weighing oT^ 
ounces troy, its diameter CJ London inches, by a fine thread on a firm 
hook, so that the distance between the hook and the centre of oscillation of 
the globe was 10| feet. I marked on the thread a point 10 feet and 1 inch 
distant from the centre of suspension and even with that point I placed a 
ruler divided into inches, by the help whereof I observed the lengths of the 
arcs described by the pendulum. Then I numbered the oscillations ia 
which the globe would lose -{- part of its motion. If the pendulum was 
drawn aside from the perpendicular to the distance of 2 inches, and thence 
let go, so that in its whole descent it described an arc of 2 inches, and in 
the first whole oscillation, compounded of the descent and subsequent 
ascent, an arc of almost 4 inches, the same in 164 oscillations lost j part 
of its motion, so as in its last ascent to describe an arc of If inches. If 
in the first descent it described an arc of 4 inches, it lost j part of its mo 
tion in 121 oscillations, so as in its last ascent to describe an arc of 3^ 
inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches, 
it lost | part of its motion in 69, 35|, 18|- 7 9| oscillations, respectively. 
Therefore the difference between the arcs described in the first descent and 
the last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, }, 1. 1, 2, 4, 8 
inches respectively. Divide those differences by the number of oscillations 
in each case, and in one mean oscillation, wherein an arc of 3 , 7-|, 15, 30 



314 THE MATHEMATICAL PRINCIPLES [BOOK Jl. 

60, 120 inches was described, the difference of the arcs described in the 
descent and subsequent ascent will be |^, ^{^ e\> T 4 r; -sji fir parts of an 
inch, respectively. But these differences in the greater oscillations are in 
the duplicate ratio of the arcs described nearly, but in lesser oscillations 
something greater than in that ratio ; and therefore (by Cor. 2, Prop. XXXI 
of this Book) the resistance of the globe, when it moves very swift, is in 
the duplicate ratio of the velocity, nearly; and when it moves slowly, 
somewhat greater than in that ratio. 

Now let V represent the greatest velocity in any oscillation, and let A, 
B, and C be given quantities, and let us suppose the difference of the arcs 

3^ 

to be AV + BV 2 + CV 2 . Since the greatest velocities are in the cycloid 
as ^ the arcs described in oscillating, and in the circle as | the chords of 
those arcs ; and therefore in equal arcs are greater in the cycloid than in 
the circle in the ratio of | the arcs to their chords ; but the times in the 
circle are greater than in the cycloid, in a reciprocal ratio of the velocity ; 
it is plain that the differences of the arcs (which are as the resistance and 
the square of the time conjunctly) are nearly the same in both curves : for 
in the cycloid those differences must be on the one hand augmented, with 
the resistance, in about the duplicate ratio of the arc to the chord, because 
of the velocity augmented in the simple ratio of the same ; and on the 
other hand diminished, with the square of the time, in the same duplicate 
ratio. Therefore to reduce these observations to the cycloid, we must take 
the same differences of the arcs as were observed in the circle, and suppose 
the greatest velocities analogous to the half, or the whole arcs, that is, to 
the numbers , 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6th cases, put 
1,4, and 1 6 for V ; and the difference of the arcs in the 2d case will become 

i 2 

* = A + B + C; in the4th case, ^- = 4A + SB + 160 ; in the 6th 

121 OOj 

case, ^ = 16A + 64B -f- 256C. These equations reduced give A = 

9? 
0,000091 6, B =-. 0,0010847, and C = 0,0029558. Therefore the difference 

of the arcs is as 0,0000916V -f 0,0010847V* + 0,0029558 V* : and there 
fore since (by Cor. Prop. XXX, applied to this case) the re.-ist;mcc of the 
globe in the middle of the arc described in oscillating, where the velocity 

is V, is to its weight as T 7 T AV -f- T \BV^ + f CV 2 to the length of the 
pendulum, if for A, B, and C you put the numbers found, the resistance of 

the globe will be to its weight as 0,0000583V + 0,0007593V* + 0,OJ22169V 2 
to the length of the pendulum between the centre of suspension and the 
ruler, that is, to 121 inches. Therefore since V in the second case repre 
sents 1, in the 4th case 4, and in the 6th case 16, the resistance will be to 
the weight of the globe, in the 2d case, as 0,0030345 to 121 ; in the 4th, as 
0,041748 to 121 ; in the 6th, as 0,61705 to 121. 



SEC. VI.] OF NATURAL PHILOSOPHY. 315 

The arc, which the point marked in the thread described in the 6th case, 

was of 120 Q^, or 119/g inches. And therefore since the radius was 

y a 

121 inches, and the length of the pendulum between the point of suspen 
sion and the centre of the globe was 126 inches, the arc which the centre of 
the globe described was 124/ T inches. Because the greatest velocity of the 
oscillating body, by reason of the resistance of the air, does not fall on the 
lowest point of the arc described, but near the middle place of the whole 
arc, this velocity will be nearly the same as if the globe in its whole descent 
in a non-resisting medium should describe 62 ^ inches, the half of that arc, 
and that in a cycloid, to which we have above reduced the motion of the 
pendulum; and therefore that velocity will be equal to that which the 
globe would acquire by falling perpendicularly from a height equal to the 
versed sine of that arc. But that versed sine in the cycloid is to that arc 
62/2 as the same arc to twice the length of the pendulum 252, and there 
fore equal to 15,278 inches. Therefore the velocity of the pendulum is the 
same which a body would acquire by falling, and in its fall describing a 
space of 15,278 inches. Therefore with such a velocity the globe meets 
with a resistance which is to its weight as 0,61705 to 121, or (if we take 
that part only of the resistance which is in the duplicate ratio of the ve- 
loc.ty) as 0,56752 to 121. 

I found, by an hydrostatical experiment, that the weight of this wooden 
globe was to the weight of a globe of water of the same magnitude as 55 
to 97: and therefore since 121 is to 213,4 in the same ratio, the resistance 
made to this globe of water, moving forwards with the above-mentioned 
velocity, will be to its weight as 0,56752 to 213,4, that is, as 1 to 376^. 
Whence since the weight of a globe of water, in the time in which the 
globe with a velocity uniformly continued describes a length of 30,556 
inches, will generate all that velocity in the falling globe, it is manifest 
that the force of resistance uniformly continued in the same time will take 
away a velocity, which will be less than the other in the ratio of 1 to 376^- , 

that is, the rr^-r part of the whole velocity. And therefore in the time 

37 VSG 

Jiat the globe, with the same velocity uniformly continued, would describe 
the length of its semi-diameter, or 3 T \ inches, it would lose the 3^42 part 
of its motion. 

I also counted the oscillations in which the pendulum lost j part of its 
motion. In the following table the upper numbers denote the length of the 
arc described in the first descent, expressed in inches and parts of an inch ; 
the middle numbers denote the length of the arc described in the last as 
cent ; and in the lowest place are the numbers of the oscillations. I give 
un account of this experiment, as being more accurate than that in which 



316 THE MATHEMATICAL PRINCIPLES [BOOK ll 

only 1 part of the motion was lost. I leave the calculation to such as are 
disposed to make it. 

First descent ... 2 4 8 16 32 64 

Last ascent . . , 1| 3 6 12 24 48 

Numb.ofoscilL . .374 272 162i 83J 41f 22| 

I afterward suspended a leaden globe of 2 inches in diameter, weighing 
26 1 ounces troy by the same thread, so that between the centre of the 
globe and the point of suspension there was an interval of 10^ feet, and 1 
counted the oscillations in which a given part of the motion was lost. The 
iirst of the following tables exhibits the number of oscillations in which J- 
part of the whole motion was lost ; the second the number of oscillations 
in which there was lost \ part of the same. 

First descent .... 1 2 4 8 16 32 64 

Last ascent .... f J 3^ 7 14 28 56 

Numb, of oscilL . . 226 228 193 140 90^ 53 30 

First descent .... 1 2 4 8 16 32 64 

Last ascent .... 1^ 3 6 12 24 4S 

Nunib. of oscill. . .510 518^ 420 318 204 12170 

Selecting in the first table the 3d, 5th, and 7th observations, and express 
ing the greatest velocities in these observations particularly by the num 
bers 1, 4, 16 respectively, and generally by the quantity V as above, there 

will come out in ihe 3d observation ~- = A + B + C, in the 5th obser- 



2 8 

vation ^ = 4A 4- 8B + 16C. in the 7th observation ^-- == 16A 4- 64B t- 
,t(j j oU 

256C. These equations reduced give A = 0,001414, B == 0,000297, C 
0,000879. And thence the resistance of the globe moving with the velocity 
V will be to its weight 26^ ounces in the same ratio as 0,0009V + 

0,000208V* + 0,000659V 2 to 121 inches, the length of the pendulum. 
And if we regard that part only of the resistance which is in the dupli 
cate ratio of the velocity, it will be to the weight of the globe as 0,000659V 2 
to 121 inches. But this part of the resistance in the first experiment was 
to the weight oi the wooden globe of 572- 7 2 ounces as 0,002217V 2 to 121 ; 
and thence the resistance of the wooden globe is to the resistance of the 
leaden one (their velocities being equal) as 57/2- i nto 0,002217 to 26 J- 
into 0,000659, that is, as 7|- to 1. The diameters of the two globes were 
6f and 2 inches, and the squares of these are to each other as 47 and 4, 
or 11-J-f and 1, nearly. Therefore the resistances of these equally swift 
globes were in less than a duplicate ratio of the diameters. But we have 
not yet considered the resistance of the thread, which was certainly very 
considerable, and ought to be subducted from the resistance of the pendu 
lums here found. I could not determine this accurately, but I found il 



SEC. VI.J OF NATURAL PHILOSOPHY. 3 1/ 

greater than a third part of the whole resistance of the lesser pendulum ; 
and thence I gathered that the resistances of the globes, when the resist 
ance of the thread is subducted, are nearly in the duplicate ratio of their 
diameters. For the ratio of 7} } to 1 , or l(H to 1 is not very 
different from the duplicate ratio of the diameters 1 L}f to I. 

Since the resistance of the thread is of less moment in greater globes, I 
tried the experiment also with a globe whose diameter was ISf inches. 
The length of the pendulum between the point of suspension and the cen 
tre of oscillation was 122| inches, and between the point of suspension and 
the knot in the thread 109| inches. The arc described by the knot at the 
first descent of the pendulum was 32 inches. The arc described by the 
same knot in the last ascent after five oscillations was 2S inches. The 
sum of the arcs, or the whole arc described in one mean oscillation, was 60 
inches. The difference of the arcs 4 inches. The y 1 ,,- part of this, or the 
difference between the descent and ascent in one mean oscillation, is f of 
an inch. Then as the radius 10 ( J| to the radius 122^, so is the whole arc 
of 60 inches described by the knot in one mean oscillation to the whole arc 
of 67} inches described by the centre of the globe in one mean oscillation ; 
and so is the difference | to a new difference 0,4475. If the length of the 
arc described were to remain, and the length of the pendulum should be 
augmented in the ratio of 126 to 122}, the time of the oscillation would 
be augmented, and the velocity of the pendulum would be diminished in 
the subduplicate of that ratio ; so that the difference 0,4475 of the arcs de 
scribed in the descent and subsequent ascent would remain. Then if the 
arc described be augmented in the ratio of 124 3 3 T to 67}, that difference 
0.4475 would be augmented in the duplicate of that ratio, and so would 
become 1,5295. These things would be so upon the supposition that the 
resistance of the pendulum were in the duplicate ratio of the velocity. 
Therefore if the pendulum describe the whole arc of 1243 3 T inches, and its 
length between the point of suspension and the centre of oscillation be 126 
inches, the difference of the arcs described in the descent and subsequent 
ascent would be 1,5295 inches. And this difference multiplied into the 
weight of the pendulous globe, which was 208 ounces, produces 318,136. 
Again ; in the pendulum above-mentioned, made of a wooden globe, when 
its centre of oscillation, being 126 inches from the point of suspension, de 
scribed the whole arc of 124 / T inches, the difference of the arcs described 

in the descent and ascent was ^^ into ^. This multiplied into the 

i/wi y^ 

weight of the globe, which was 57-2 7 2 ounces, produces 49,396. But I mul 
tiply these differences into the weights of the globes, in order to find their 
resistances. For the differences arise from the resistances, and are as the 
resistances directly and the weights inversely. Therefore the resistances 
are as the numbers 318,136 and 49,396. But that part of the resistance 



31 S THE MATHEMATICAL PRINCIPLES [BOOK 1L 

of the lesser globe, which is in the duplicate ratio of the velocity, was to 
the whole resistance as 0,56752 to- 0,61675, that is, as 45,453 to 49,396 ; 
whereas that part of the resistance of the greater globe is almost equal to 
its whole resistance ; and so those parts are nearly as 318,136 and 45,453, 
that is, as 7 and 1. But the diameters of the globes are 18f and 6| ; and 
their squares 351 T 9 and 47 J are as 7,438 and 1, that is, as the resistances 
of the globes 7 and 1, nearly. The difference of these ratios is scarce 
greater than may arise from the resistance of the thread. Therefore those 
parts of the resistances which are, when the globes are equal, as the squares 
of the velocities, are also, when the velocities are equal, as the squares of 
the diameters of the globes. 

But the greatest of the globes I used in these experiments was not per 
fectly spherical, and therefore in this calculation I have, for brevity s sake, 
neglected some little niceties ; being not very solicitous for an accurate 
calculus in an experiment that was not very accurate. So that I could 
wish that these experiments were tried again with other globes, of a larger 
size, more in number, and more accurately formed ; since the demonstra 
tion of a vacuum depends thereon. If the globes be taken in a geometrical 
proportion, as suppose whose diameters are 4, 8, 16, 32 inches; one may 
collect from the progression observed in the experiments what would hap 
pen if the globes were still larger. 

In order to compare the resistances of different fluids with each other, 1 
made the following trials. I procured a wooden vessel 4 feet long, 1 foot 
broad, and 1 foot high. This vessel, being uncovered, 1 filled with spring 
water, and, having immersed pendulums therein, I made them oscillate in 
the water. And I found that a leaden globe weighing 166| ounces, and in 
diameter 3f inches, moved therein as it is set down in the following table ; 
the length of the pendulum from the point of suspension to a certain 
point marked in the thread being 126 inches, and to the centre of oscilla 
tion 134 f inches. 
The arc described in } 

the first descent, by 

a point marked in \ 64 . 32 . 16 . $ . 4 . 2 . 1 . . J 

the thread was \ 

inches. 
The arc described in ) 

the last ascent was V 48 . 24 . 12 . 6 . 3 . 1| . . f . T \ 

inches. \ 

The difference of the 

arcs, proportional 

to the motion lost, 

was inches. 
The number of the os 
cillations in water. 
The number of the os 
cillations in air. 



16 



. li . 3 . 7 . lH.12f.13j 



85i . 287 . 535 



SEC. VI.] OF NATURAL PHILOSOPHY. 319 

In the experiments of the 4th column there were equal motions lost in 
535 oscillations made in the air, and If in water. The oscillations in the 
air were indeed a little swifter than those in the water. But if the oscil 
lations in the water were accelerated in such a ratio that the motions of 
the pendulums might be equally swift in both mediums, there would be 
still the same number 1 j of oscillations in the water, and by these the 
same quantity of motion would be lost as before ; because the resistance i> 
increased, and the square of the time diminished in the same duplicate ra 
tio. The pendulums, therefore, being of equal velocities, there were equal 
motions lost in 535 oscillations in the air, and 1} in the water; and there 
fore the resistance of the pendulum in the water is to its resistance in the 
air as 535 to 1 }. This is the proportion of the whole resistances in the 
case of the 4th column. 

Now let AV + CV 2 represent the difference of the arcs described in the 
descent and subsequent ascent by the globe moving in air with the greatest 
velocity V ; and since the greatest velocity is in the case of the 4th column 
to the greatest velocity in the case of the 1st column as 1 to 8 ; and that 
difference of the arcs in the case of the 4th column to the difference in the 

2 16 

case of the 1st column as ^ to 7, or as 86 J to 4280 ; put in these 



cases 1 and 8 for the velocities, and 85 1 and 4280 for the differences of 
the arcs, and A + C will be S5|, and 8A -f 640 == 4280 or A + SC 
= 535 ; and then by reducing these equations, there will come out TC = 
449^ and C = 64 T \ and A = 21f ; and therefore the resistance, which is 
as T VAV + fCV 2 , will become as 13 T 6 T V + 48/^Y 2 . Therefore in the 
case of the 4th column, where the velocity was 1, the whole resistance is to 
its part proportional to the square of the velocity as 13 T 6 T + 48/ F or 
61 }f to 48/e ; and therefore the resistance of the pendulum in water is to 
that part of the resistance in air, which is proportional to the square of the 
velocity, and which in swift motions is the only part that deserves consid 
eration, as 61}^ to 4S/g and 535 to 1} conjunctly, that is, as 571 to 1. 
If the whole thread of the pendulum oscillating in the water had been im 
mersed, its resistance would have been still greater ; so that the resistance 
of the pendulum oscillating in the water, that is, that part which is pro 
portional to the square of the velocity, and which only needs to be consid 
ered in swift bodies, is to the resistance of the same whole pendulum, oscil 
lating in air with the same velocity, as about 850 to 1, that is as, the den 
sity of water to the density of air, nearly. 

In this calculation we ought also to have taken in that part of the re 
sistance of the pendulum in the water which was as the square of the ve 
locity ; but I found (which will perhaps seem strange) that the resistance 
in the water was augmented in more than a duplicate ratio of the velocity. 
In searching after the cause, I thought upon this, that the vessel was toe 



320 THE MATHEMATICAL PRINCIPLES [BOOK II. 

narrow for the magnitude of the pendulous globe, and by its narrowness 
obstructed the motion of the water as it yielded to the oscillating globe. 
For when I immersed a pendulous globe, whose diameter was one inch only, 
the resistance was augmented nearly in a duplicate ratio of the velocity, 
I tried this by making a pendulum of two globes, of which the lesser and 
lower oscillated in the water, and the greater and higher was fastened to 
the thread just above the water, and, by oscillating in the air, assisted the 
motion of the pendulum, and continued it longer. The experiments made 
by this contrivance proved according to the following table. 
Arc descr. in first descent . .16.8. 4. 2.1.1.1 
Arc descr. in last ascent . . 12 . 6 . 3 . li . J . | . T 3 F 
Dif. of arcs, proport. to 1 . pi i 

motion lost $ T r T* 

Number of oscillations... 3f . 6j . 12^. 211 . 34 . 53 . 62) 

In comparing the resistances of the mediums with each other, I also 
caused iron pendulums to oscillate in quicksilver. The length of the iron 
wire was about 3 feet, and the diameter of the pendulous globe about i of 
an inch. To the wire, just above the quicksilver, there was fixed another 
leaden globe of a bigness sufficient to continue the motion of the pendulum 
for some time. Then a vessel, that would hold about 3 pounds of quick 
silver, was filled by turns with quicksilver and common water, that, by 
making the pendulum oscillate successively in these two different fluids, I 
might find the proportion of their resistances ; and the resistance of the 
quicksilver proved to be to the resistance of water as about 13 or 14 to 1 ; 
that is. as the density of quicksilver to the density of water. When I made 
use of a pendulous globe something bigger, as of one whose diameter was 
about ^ or | of an inch, the resistance of the quicksilver proved to be to 
the resistance of the water as about 12 or 10 to 1. But the former experi 
ment is more to be relied on, because in the latter the vessel was too nar 
row in proportion to the magnitude of the immersed globe; for the vessel 
ought to have been enlarged together with the globe. I intended to have 
repeated these experiments with larger vessels, and in melted metals, and 
other liquors both cold and hot ; but I had not leisure to try all: and be 
sides, from what is already described, it appears sufficiently that the resist 
ance of bodies moving swiftly is nearly proportional to the densities of 
the fluids in which they move. I do not say accurately ; for more tena 
cious fluids, of equal density, will undoubtedly resist more than those that 
are more liquid ; as cold oil more than warm, warm oil more than rain 
water, and water more than spirit of wine. But in liquors, which are sen 
sibly fluid enough, as in air, in salt and fresh water, in spirit of wine, of 
turpentine, and salts, in oil cleared of its fseces by distillation and warmed, 
in oil of vitriol, and in mercury, and melted metals, and any other such 
like, that are fluid enough to retaia for some time the motion impressed 



SEC. VI.J OF NATURAL PHILOSOPHY. 321 

upon them by the agitation of the vessel, and which being poured out are 
easily resolved into drops, I doubt not but the rule already laid down may 
be accurate enough, especially if the experiments be made with larger 
pendulous bodies and more swiftly moved. 

Lastly, since it is the opinion of some that there is a certain ^ethereal 
medium extremely rare and subtile, which freely pervades the pores of all 
bodies ; and from such a medium, so pervading the pores of bodies, some re 
sistance must needs arise; in order to try whether the resistance, which w r e 
experience in bodies in motion, be made upon their outward superficies only, 
or whether their internal parts meet with any considerable resistance upon 
their superficies, I thought of the following experiment I suspended a 
round deal box by a thread 11 feet long, on a steel hook, by means of a ring 
of the s-ime metal, so as to make a pendulum of the aforesaid length. The 
hook had a sharp hollow r edge on its upper part, so that the upper arc of 
the ring pressing on the edge might move the more freely ; and the thread 
was fastened to the lower arc of the ring. The pendulum being thus pre 
pared, I drew it aside from the perpendicular to the distance of about 6 
feet, and that in a plane perpendicular to the edge of the hook, lest the 
ring, while the pendulum oscillated, should slide to and fro on the edge of 
the hook : for the point of suspension, in which the ring touches the hook, 
ought to remain immovable. I therefore accurately noted the place to 
which the pendulum was brought, and letting it go, I marked three other 
places, to which it returned at the end of the 1st, 2d, and 3d oscillation. 
This I often repeated, that I might find those places as accurately as pos 
sible. Then I filled the box with lead and other heavy metals that were 
near at hand. But, first, I weighed the box when empty, and that pnrt of 
the thread that went round it, and half the remaining part, extended be 
tween the hook and the suspended box ; for the thread so extended always 
acts upon the pendulum, when drawn aside from the perpendicular, with half 
its weight. To this weight I added the weight of the air contained in the 
box And this whole weight was about -fj of the weight of the box when 
filled w r ith the metals. Then because the box when full of the metals, by ex 
tending the thread with its weight, increased the length of the pendulum, 
f shortened the thread so as to make the length of the pendulum, when os 
cillating, the same as before. Then drawing aside the pendulum to the 
place first marked, and letting it go, I reckoned about 77 oscillations before 
the box returned to the second mark, and as many afterwards before it came 
to the third mark, and as many after that before it came to the fourth 
xnark. From whence I conclude that the whole resistance of the box, when 
full, had not a greater proportion to the resistance of the box, when empty, 
than 78 to 77. For if their resistances were equal, the box, when full, by 
reason of its vis insita, which was 78 times greater than the vis tfuritoof 
the same when empty, ought to have continued its oscillating motion so 

21 



322 THE MATHEMATICAL PRINCIPLES | BOOK II. 

much the longer, and therefore to have returned to those marks at the end 
of 78 oscillations. But it returned to them at the end of 77 oscillations. 

Let, therefore, A represent the resistance of the box upon its external 
superficies, and B the resistance of the empty box on its internal superficies ; 
and if the resistances to the internal parts of bodies equally swift be as the 
matter, or the number of particles that are resisted, then 78B will be the 
resistance made to the internal parts of the box, when full ; and therefore 
the whole resistance A + B of the empty box will be to the whole resist 
ance A + 7SB of the full box as 77 to 78, and, by division, A + B to 77B 
as 77 to 1 ; and thence A + B to B as 77 X 77 to 1, and, by division 
again, A to B as 5928 to 1. Therefore the resistance of the empty box in 
its internal parts will be above 5000 times less than the resistance on its 
external superficies. This reasoning depends upon the supposition that the 
greater resistance of the full box arises not from any other latent cause, 
but only from the action of some subtile fluid upon the included metal. 

This experiment is related by memory, the paper being lost in which I 
had described it ; so that I have been obliged to omit some fractional parts, 
which are slipt out of my memory ; and I have no leisure to try it again. 
The first time I made it, the hook being weak, the full box was retarded 
sooner. The cause I found to be, that the hook was not strong enough to 
bear the weight of the box : so that, as it oscillated to and fro, the hook 
was bent sometimes this and sometimes that way. I therefore procured a 
hook of sufficient strength, so that the point of suspension might remain 
unmoved, and then all things happened as is above described. 



SEC. VI I.] OF NATURAL PHILOSOPHY. 323 

SECTION VII. 

Of the, motion of fluids, and the resistance made to projected bodies. 

PROPOSITION XXXII. THEOREM XXVI. 

Suppose two similar systems of bodies consisting of an equal number of 
particles, and let the correspondent particles be similar and propor 
tional, each in, one system to each in the other, and have a like situa 
tion among themselves, and the same given ratio of density to each 
other ; and let them begin to move anwng themselves in proportional 
times, and with like motions (that is, those in one system among one 
another, and those in the other among one another). And if the par 
ticles that are in the same system do not touch otte another, except ir 
the moments of reflexion ; nor attract, nor repel each other, except with 
accelerativeforc.es that are as the diameters of the correspondent parti 
cles inversely, and the squares of the velocities directly ; I say, that the 
particles of those systems will continue to move among themselves wit It 
like motions and in proportional times. 

Like bodies in like situations are said to be moved among themselves 
with like motions and in proportional times, when their situations at the 
end of those times are always found alike in respect of each other ; as sup 
pose we compare the particles in one system with the correspondent parti 
cles in the other. Hence the times will be proportional, in which similar 
and proportional parts of similar figures will be described by correspondent 
particles. Therefore if we suppose two systems of this kind; the corre 
spondent particles, by reason of the similitude of the motions at their 
beginning, will continue to be moved with like motions, so long as they 
move without meeting one another ; for if they are acted on by no forces, 
they will go on uniformly in right lines, by the 1st Law. But if they do 
agitate one another with some certain forces, and those forces are as the 
diameters of the correspondent particles inversely and the squares of the 
velocities directly, then, because the particles are in like situations, and 
their forces are proportional, the whole forces with which correspondent 
particles are agitated, and which are compounded of each of the agitating 
forces (by Corol. 2 of the Laws), will have like directions, and have the 
same effect as if they respected centres placed alike among the particles ; 
and those whole forces will be to each other as the several forces which 
compose them, that is, as the diameters of the correspondent particles in 
versely, and the squares of the velocities directly : and therefore will cans** 



3^4 THE MATHEMATICAL PRINCIPLES [BOOK 11. 

correspondent particles to continue to describe like figures. These things 
will be so (by Cor. 1 and S, Prop. IV. ; Book 1), if those centres are at rest 
but if they are moved, yet by reason of the similitude of the translations, 
their situations among the particles of the system will remain similar , so 
that the changes introduced into the figures described by the particles will 
still be similar. So that the motions of correspondent and similar par 
ticles will continue similar till their first meeting with each other ; and 
thence will arise similar collisions, and similar reflexions; which will again 
beget similar motions of the particles among themselves (by what was just 
now shown), till they mutually fall upon one another again, and so on ad 
infinitum. 

COR. 1. Hence if any two bodies, which are similar and in like situations 
to the correspondent particles of the systems, begin to move amongst them 
in like manner and in proportional times, and their magnitudes and densi 
ties be to each other as the magnitudes and densities of the corresponding 
particles, these bodies will continue to be moved in like manner and in 
proportional times: for the case of the greater parts of both systems and of 
the particles is the very same. 

COR. 2. And if all the similar and similarly situated parts of both sys 
tems be at rest among themselves ; and two of them, which are greater than 
the rest, and mutually correspondent in both systems, begin to move in 
lines alike posited, with any similar motion whatsoever, they will excite 
similar motions in the rest of the parts of the systems, and will continue 
to move among those parts in like manner and in proportional times ; and 
will therefore describe spaces proportional to their diameters. 



PROPOSITION XXXIII. THEOREM XXVII. 

The same things biting supposed, I say, that the greater parts of the 
systems are resisted in a ratio compounded of the duplicate ratio of 
their velocities, and the duplicate ratio of their diameters, and Ihe sim 
ple ratio of the density of the parts of the systems. 
For the resistance arises partly from the centripetal or centrifugal, forces 
with which the particles of the system mutually act on each other, partly 
from the collisions and reflexions of the particles and the greater parts. 
The resistances of the first kind are to each other as the whole motive 
forces from which they arise, that is, as the whole accelerative forces and 
the quantities of matter in corresponding parts ; that is (by the sup 
position), as the squares of the velocities directly, and the distances of the 
corresponding particles inversely, and the quantities of matter in the cor 
respondent parts directly : and therefore since the distances of the parti 
cles in one system are to the correspondent distances of the particles of the 
;ther S3 the diameter of one particle or part in *he former system to the 



SEC. VII.] OF NATURAL PHILOSOPHY. C>2" 

diameter of the correspondent particle or part in the other, and since the 
quantities of matter are as the densities of the parts and the cubes of the 
diameters ; the resistances arc to each other as the squares of the velocities 
and the squares of the diameters and the densities of the parts of the sys 
tems. Q.E.D. The resistances of the latter sort are as the number of 
sorrespondent reflexions and the forces of those reflexions conjunctly ; but 
the number of the reflexions are to each other as the velocities of the cor 
responding parts directly and the spaces between their reflexions inversely. 
And the forces of the reflexions are as the velocities and the magnitudes 
and the densities of the corresponding parts conjunctly ; that is, as the ve 
locities and the cubes of the diameters and the densities of the parts. And, 
joining all these ratios, the resistances of the corresponding parts are to 
each other as the squares of the velocities and the squares of the diameters 
and the densities of the parts conjunctly. Q.E.T). 

COR. 1. Therefore if those systems are two elastic fluids, like our air, 
and their parts are at rest among themselves ; and two similar bodies pro 
portional in magnitude and density to the parts of the fluids, and similarly 
gituated among those parts, be any how projected in the direction of lines 
similarly posited ; and the accelerative forces with which the particles of 
the fluids mutually act upon each other are as the diameters of the bodies 
projected inversely and the squares of their velocities directly ; those bodies 
will excite similar motions in the fluids in proportional times, and will de 
scribe similar spaces and proportional to their diameters. 

COR. 2. Therefore in the same fluid a projected body that moves swiftly 
meets with a resistance that is, in the duplicate ratio of its velocity, nearly. 
For if the forces with which distant particles act mutually upon one 
another should be augmented in the duplicate ratio of the velocity, the 
projected body would be resisted in the same duplicate ratio accurately ; 
and therefore in a medium, whose parts when at a distance do not act mu 
tually with any force on one another, the resistance is in the duplicate ra 
tio of the velocity accurately. Let there be, therefore, three mediums A, 
B, C, consisting of similar and equal parts regularly disposed at equal 
distances. Let the parts of the mediums A and B recede from each other 
with forces that are among themselves as T and V ; and let the parts of 
the medium C be entirely destitute of any such forces. And if four equal 
bodies D, E, P 7 G, move in these mediums, the two first D and E in the 
two first A and B, and the other two P and G in the third C ; and if the 
velocity of the body D be to the velocity of the body E, and the velocity 
of the body P to the velocity of the body G, in the subduplicate ratio of 
the force T to the force V ; the resistance of the body D to the resistance 
of the body E, and the resistance of the body P to the resistance of the 
body G, will be in the duplicate ratio of the velocities ; and therefore the 
resistance of the body D will be to the resistance of the body P as the re- 



326 THE MATHEMATICAL PRINCIPLES [BOOK II 

sistance of the body E to the resistance of the body G. Let the bodies 1) 
and F be equally swift, as also the bodies E and G ; and, augmenting the 
velocities of the^bodies D arid F in any ratio, and diminishing the forces 
of the particles of the medium B in the duplicate of the same ratio, the 
medium B will approach to the form and condition of the medium C at 
pleasure ; and therefore the resistances of the equal and equally swift 
bodies E and G in these mediums will perpetually approach to equality 
so that their difference will at last become less than any given. There 
fore since the resistances of the bodies D and F are to each other as the 
resistances of the bodies E and G, those will also in like manner approach 
to the ratio of equality. Therefore the bodies 1) and F, when they move 
with very great swiftness, meet with resistances very nearly equal; and 
therefore since the resistance of the body F is in a duplicate ratio of the 
velocity, the resistance of the body D will be nearly in the same ratio. 

Con. 3. The resistance of a body moving very swift in an elastic fluid 
is almost the same as if the parts of the fluid were destitute of their cen 
trifugal forces, and did not fly from each other; if so be that the elasti 
city of the fluid arise from the centrifugal forces of the particles, and the 
velocity be so great as not to allow the particles time enough to act. 

COR. 4. Therefore, since the resistances of similar and equally swift 
bodies, in a medium whose distant parts do not fly from each other, are as 
the squares of the diameters, the resistances made to bodies moving with 
very great and equal velocities in an elastic fluid will be as the squares of 
the diameters, nearly. 

COR. 5. And since similar, equal, and equally swift bodies, moving 
through mediums of the same density, whose particles do not fly from each 
other mutually, will strike against an equal quantity of matter in equal 
times, whether the particles of which the medium consists be more and 
smaller, or fewer and greater, and therefore impress on that matter an equal 
quantity of motion, and in return (by the 3d Law of Motion) suffer an 
equal re-action from the same, that is, are equally resisted ; it is manifest, 
also, that in elastic fluids of the same density, when the bodies move with 
extreme swiftness, their resistances are nearly equal, whether the fluids 
consist of gross parts, or of parts ever so subtile. For the resistance of 
projectiles moving with exceedingly great celerities is not much diminished 
by the subtilty of the medium. 

COR. G. All these things are so in fluids whose elastic force takes its rise 
from the centrifugal forces of the particles. But if that force arise from 
some other cause, as from the expansion of the particles after the manner 
of wool, or the boughs of trees, or any other cause, by which the particles 
are hindered from moving freely among themselves, the resistance, by 
reason of the lesser fluidity of the medium, will be greater than in the 
Corollaries above. 



SEC. VII. 



OF NATURAL PHILOSOPHY. 



32? 



K 




L, P 



O 



PROPOSITION XXXIV. THEOREM XXV1I1. 

If iu a rare medium, consisting of equal particles freely disposed at 
equal distances from each other, a globe and a cylinder described on 
equal diameters move with equal velocities in the. direction of the axis 
of the cylinder, the resistance of the globe ivill be but half so great an 
that of the cylinder. 
For since the action of the medi 
um upon the body is the same (by 
Cor. 5 of the Laws) whether the body 
move in a quiescent medium, or 
whether the particles of the medium 
impinge with the same velocity upon 
the quiescent body, let us consider 
the body as if it were quiescent, and 
see with what force it would be im- 
pelled by the moving medium. Let, therefore, ABKI represent a spherical 
body described from the centre C with the semi-diameter CA, and let the 
particles of the medium impinge with a given velocity upon that spherical 
body in the directions of right lines parallel to AC : and let FB be one of 
those right lines. In FB take LB equal to the semi-diameter CB, and 
draw BI) touching the sphere in B. Upon KG and BD let fall the per 
pendiculars BE, LD ; and the force with which a particle of the medium, 
impinging on the globe obliquely in the direction FB, would strike the 
globe in B, will be to the force with which the same particle, meeting the 
cylinder ONGQ, described about the globe with the axis ACI, would strike 
it perpendicularly in b, as LD to LB, or BE to BC. Again ; the efficacy 
of this force to move the globe, according to the direction of its incidence 
FB or AC, is to the efficacy of the same to move the globe, according to 
the direction of its determination, that is, in the direction of the right line 
BC in which it impels the globe directly, as BE to BC. And, joining 
these ratios, the efficacy of a particle, falling upon the globe obliquely in 
the direction of the right line FB y to move the globe in the direction of its 
incidence, is to the efficacy of the same particle falling in the same line 
perpendicularly on the cylinder, to move it in the same direction, as BE 2 
to BC 3 . Therefore if in 6E, which is perpendicular to the circular base of 
the cylinder NAO, and equal to the radius AC, we take H equal to 

BE a 

- ; then 6H will be to 6E as the effect of the particle upon the globe t<? 
\~i\j 

the effect of the particle upon the cylinder. Arid therefore the solid which 
is formed by all the right lines 6H will be to the solid formed by all the 
right lines />E as the effect of all the particles upon the globe to the effect 
of all the particles upon the cylinder. But the former of these solids is a 



328 



THE MATHEiAlATICAL PRINCIPLES 



[BooK li. 



paraboloid whose vertex is C, its axis CA, and latus rectum CA, and the 
latter solid is a cylinder circumscribing the paraboloid ; and it is knowr 
that a paraboloid is half its circumscribed cylinder. Therefore the whole 
force of the medium upon the globe is half of the entire force of the same 
upon the cylinder. And therefore if the particles of the medium are at 
rest, and the cylinder and globe move with equal velocities, the resistance 
of the globe will be half the resistance of the cylinder. Q.E.D. 

SCHOLIUM. 

By the same method other figures may be compared together as to their 
resistance; and those may be found which are most apt to continue their 
motions in resisting mediums. As if upon the circular base CEBH from 
the centre O, with thy radius OC, and the altitude OD, one would construct 
a frustum CBGF of a cone, which should meet with less resistance than 
any other frustum constructed with the same base and altitude, and going 
forwards towards D in the direction of its axis : bisect the altitude OD in 
U,, and produce OQ, to S, so that QS may be equal to Q,C, and S will be 
the vertex of the cone whose frustum is sought. 
r 




J 




Whence, by the bye, since the angle CSB is always acute, it follows, that, 
if the solid ADBE be generated by the convolution of an elliptical or oval 
figure ADBE about its axis AB, and the generating figure be touched by- 
three right lines FG, GH, HI, in the points F, B, and I, so that GH shall 
be perpendicular to the axis in the point of contact B, arid FG, HI may be 
inclined to GH in the angles FGB, BHI of 135 degrees: the solid arising 
from the convolution of the figure ADFGH1E about the same axis AB 
will be less resisted than the former solid; if so be that both move forward 
in the direction of their axis AB, and that the extremity B of each go 
foremost. Which Proposition I conceive may be of use in the building of 
ships. 

If the figure DNFG be such a curve, that if, from any point thereof, as 
N, the perpendicular NM be let fall on the axis AB, and from the given 
point G there be drawn the right line GR parallel to a right line touching 
the figure in N, and cutting the axis produced in R, MN becomes to GR 
as GR, 3 to 4BR X GB 2 , the solid described, by the revolution of this figure 



SEC. Vll.J OF NATURAL PHILOSOPHY. 32S 

about its axis AB, moving in the before-mentioned rare medium from A 
towards B, will be less resisted than any other circular solid whatsoever, 
described of the same length and breadth. 

PROPOSITION XXXV. PROBLEM VII. 

If a rare medium consist of very small quiescent particles of equal mag 
nitudes, and freely disposed at equal distances from one another : to 
jind the resistance of a globe moving uniformly forward in this 
medium. 

CASE 1. Let a cylinder described with the same diameter and altitude be 
conceived to go forward with the same velocity in the direction of its axis 
through the same medium ; and let us suppose that the particles of the 
medium, on which the globe or cylinder falls, fly back with as great a force 
of reflexion as possible. Then since the resistance of the globe (by the last 
Proposition) is but half the resistance of the cylinder, and since the globe 
is to the cylinder as 2 to 3, and since the cylinder by falling perpendicu 
larly on the particles, and reflecting them with the utmost force, commu 
nicates to them a velocity double to its own; it follows that the cylinder. 
in moving forward uniformly half the length of its axis, will communicate 
a motion to the particles which is to the whole motion of the cylinder as 
the density of the medium to the density of the cylinder ; and that the 
globe, in the time it describes one length of its diameter in moving uni 
formly forward, will communicate the same motion to the particles ; and 
in the time that it describes tw r o thirds of its diameter, will communicate 
a motion to the particles which is to the whole motion of the globe as the 
density of the medium to the density of the globe. Arid therefore the 
globe meets with a resistance, which is to the force by which its whole mo 
tion may be either taken away or generated in the time in which it de 
scribes two thirds of its diameter moving uniformly forward, as the den 
sity of the medium to the density of the globe. 

CASE 2. Let us suppose that the particles of the medium incident on 
the globe or cylinder are not reflected ; and then the cylinder falling per 
pendicularly on the particles will communicate its own simple velocity to 
them, and therefore meets a resistance but half so great as in the former 
case, and the globe also meets with a resistance but half so great. 

CASE 3. Let us suppose the particles of the medium to fly back from 
the globe with a force which is neither the greatest, nor yet none at all, but 
with a certain mean force ; then the resistance of the globe will be in the 
same mean ratio between the resistance in the first case and the resistance 
in the second. Q.E.I. 

COR. 1. Hence if the globe and the particles are infinitely hard, and 
destitute of all elastic force, and therefore of all force of reflexion ; thf 
resistance of the globe will be to the force by which its whole motion may 




330 THE MATHEMATICAL PRINCIPLES [BOOK I) 

be destroyed or generated, in the time that the globe describes four third 
parts of its diameter, as the density of the medium to the density of the 
^lobe. 

Con. 2. The resistance of the globe, cceteris paribus, is in the duplicate 
ratio of the velocity. 

CUR. 3. The resistance of the globe, cocteris paribus, is in the duplicate 
ratio of the diameter. 

COR. 4. The resistance of the globe is, cceteris paribus, as the density of 
the medium. 

COR, 5. The resistance of the globe is in a ratio compounded of the du 
plicate ratio of the velocity, arid the duplicate ratio of the diameter, and 
the ratio of the density of the medium. 

COR. 6. The motion of the globe and its re 
sistance may be thus expounded Let AB be the 
time in which the globe may, by its resistance 
uniformly continued, lose its whole motion. 
Erect AD, BC perpendicular to AB. J ,et BC be 
that whole motion, and through the point C, the 
asymptotes being AD, AB, describe the hyperbola 
CF. Produce AB to any point E. Erect the perpendicular EF meeting 
the hyperbola in F. Complete the parallelogram CBEG, and draw AF 
meeting BC in H. Then if the globe in any time BE, with its first mo 
tion BC uniformly continued, describes in a non-resisting medium the space 
CBEG expounded by the area of the parallelogram, the same in a resisting 
medium will describe the space CBEF expounded by the area of the hv- 
perbola; and its motion at the end of that time will be expounded by EF, 
the ordinate of the hyperbola, there being lost of its motion the part FG. 
And its resistance at the end of the same time will be expounded by the 
length BH, there being lost of its resistance the part CH. All these things 
appear by Cor. 1 and 3, Prop. V., Book II. 

COR. 7. Hence if the globe in the time T by the resistance R uniformly 
continued lose its whole motion M, the same globe in the time t in a 
resisting medium, wherein the resistance R decreases in a duplicate 

/M 

ratio of the velocity, will lose out of its motion M the part ,.i the 

TM 

part rn . ; remaining ; and will describe a space which is to the space de 
scribed in the same time t, with the uniform motion M, as the logarithm of 

T + t 
the number ^. multiplied by the number 2,302585092994 is to the 

number ^ because the hyperbolic area BCFE is to the rectangle BCGE 
in that proportion. 



SEC. VII.] 



OF NATURAL PHILOSOPHY. 



331 



SCHOLIUM. 

I have exhibited in this Proposition the resistance and retardation of 
spherical projectiles in mediums that are not continued, and shewn that 
this resistance is to the force by which the whole motion of the globe may be 
destroyed or produced in the time in which the globe can describe two thirds 
of its diameter, with a velocity uniformly continued, as the density of the 
medium to the density of the globe, if so be the globe and the particles of 
the medium be perfectly elastic, and are endued with the utmost force of 
reflexion ; and that this force, where the globe and particles of the medium 
are infinitely hard and void of any reflecting force, is diminished one half. 
But in continued mediums, as water, hot oil, and quicksilver, the globe as 
it passes through them does not immediately strike against all the parti 
cles of the fluid that generate the resistance made to it, but presses only 
the particles that lie next to it, which press the particles beyond, which 
press other particles, and so on ; and in these mediums the resistance is di 
minished one other half. A globe in these extremely fluid mediums meets 
with a resistance that is to the force by which its whole motion may be 
destroyed or generated in the time wherein it can describe, with that mo 
tion uniformly continued, eight third parts of its diameter, as the density 
of the medium to the density of the globe. This I shall endeavour to shew 
in what follows. 



PROPOSITION XXXVI. PROBLEM VIII. 

To define the motion of water running out of a cylindrical vessel through 

a hole made at the bottom. 

Let AC D B be a cylindrical vessel, AB the mouth p = Q: 

of it, CD the bottom p irallel to the horizon, EF a 
circular hole in the middle of the bottom, G the 
c-?ritre of the hole, and GH the axis of the cylin- K j 
cler perpendicular to the horizon. And suppose a 
cylinder of ice APQ,B to be of the same breadth 
with the cavity of the vessel, and to have the same 
axis, and to descend perpetually with an uniform 
motion, and that its parts, as soon as they touch the 
superficies AB, dissolve into water, and flow 
( wn by their weight into the vessel, and in their 
fall compose the cataract or column of water 
ABNFEM, passing through the hole EF, and filling up the same exactly. 
Let the uniform velocity of the descending ice and of the contiguous water 
in the circle AB be that which the water would acquire by falling through 
the space IH ; and let IH and HG lie in the same right line ; and through 




332 THE MATHEMATICAL PRINCIPLES [BOOK Jl 

the point I let there be drawn the right line KL parallel to the horizon 
and meeting the ice on both the sides thereof in K and L. Then the ve 
locity of the water running out at the hole EF will be the same that it 
would acquire by falling from I through the space IG. Therefore, by 
Galih cJ s Theorems, IG will be to IH in the duplicate ratio of the velo 
city of the water that runs out at the hole to the velocity of the w r ater in 
the circle AB, that is, in the duplicate ratio of the circle AB to the circle 
EF ; those circles being reciprocally as the velocities of the water which 
in the same time and in equal quantities passes severally through each of 
them, and completely fills them both. We are now considering the velo 
city with which the water tends to the plane of the horizon. But the mo 
tion parallel to the same, by which the parts of the falling water approach to 
each other, is not here taken notice of; since it is neither produced by 
gravity, nor at all changes the motion perpendicular to the horizon which the 
gravity produces. We suppose, indeed, that the parts of the water cohere 
a little, that by their cohesion they may in falling approach to each othei 
with motions parallel to the horizon in order to form one single cataract, 
and to prevent their being divided into several : but the motion parallel to 
the horizon arising from this cohesion does not come under our present 
consideration. 

CASE 1. Conceive now the w^hole cavity in the vessel, w r hich encompasses 
the falling water ABNFEM, to be full of ice, so that the water may pass 
through the ice as through a funnel. Then if the water pass very near to 
the ice only, without touching it; or, which is the same tiling, if by rea 
son of the perfect smoothness of the surface of the ice, the water, though 
touching it. glides over it w r ith the utmost freedom, and without the le-ast 
resistance; the water will run through the hole EF with the same velocity 
as before, and the whole weight of the column of water ABNFEM will be 
all taken up as before in forcing out the water, and the bottom of the vessel 
will sustain the weight of the ice encompassing that column. 

Let now the ice in the vessel dissolve into water ; yet will the efflux of 
the water remain, as to its velocity, the same as before. It will not be 
less, because the ice now dissolved will endeavour to descend ; it will not 
be greater, because the ice. now become water, cannot descend without hin 
dering the descent of other water equal to its own descent. The same force 
ought always to generate the same velocity in the effluent water. 

But the hole at the bottom of the vessel, by reason of the oblique mo 
tions of the particles of the effluent water, must be a little greater than before, 
For now the particles of the water do not all of them pass through the 
hole perpendicularly, but, flowing down on all parts from the sides of the 
vessel, and converging towards the hole, pass through it with oblique mo 
tions : r,r,d in tending downwards meet in a stream whose diameter is a little 
smaller below the hole than at the hole itself : its diameter being to the 



SEC. V1L! 



OF NATURAL PHILOSOPHY. 



333 



diameter of the hole as 5 to 6, or as 5^ to 6|, very nearly, if I took the 
measures of those diameters right. I procured a very thin flat plate, hav 
ing a hole pierced in the middle, the diameter of the circular hole being 
f parts of an inch. And that the stream of running waters might not be 
accelerated in falling, and by that acceleration become narrower, I fixed 
this plate not to the bottom, but to the side of the vessel, so us to make the 
water go out in the direction of a line parallel to the horizon. Then, when 
the vessel was full of water, I opened the hole to let it run out ; and the 
diameter of the stream, measured with great accuracy at the distance of 
about half an inch from the hole, was f J- of an inch. Therefore the di 
ameter of this circular hole was to the diameter of the stream very nearly 
as 25 to 21. So that the water in passing through the hole converges on 
all sides, and, after it has run out of the vessel, becomes smaller by converg 
ing in that manner, and by becoming smaller is accelerated till it comes to 
the distance of half an inch from the hole, and at that distance flows in a 
smaller stream and with greater celerity than in the hole itself, and this 
in the ratio of 25 X 25 to 21 X 21, or 17 to 12, very nearly ; that is, in 
about the subdaplicate ratio of 2 to 1. Now it is certain from experiments, 
that the quantity of water running out in a given time through a circular 
hole made in the bottom of a vessel is equal to the quantity, which, flow 
ing with the aforesaid velocity, would run out in the same time through 
another circular hole, whose diameter is to the diameter of the former as 
21 to 25. And therefore that running water in passing through the 
hole itself has a velocity downwards equal to that which a heavy body 
would acquire in falling through half the height of the stagnant water in 
the vessel, nearly. But, then, after it has run out, it is still accelerated by 
converging, till it arrives at a distance from the hole that is nearly equal to 
its diameter, and acquires a velocity greater than the other in about the 
subduplicate ratio of 2 to 1 ; which velocity a heavy body would nearly 
acquire by falling through the whole height of the stagnant water in the 
vessel. 

Therefore in what follows let the diameter of 
the stream be represented by that lesser hole which 
we called EF. And imagine another plane VW 
above the hole EF, and parallel to the plane there 
of, to be placed at a distance equal to the diame 
ter of the same hole, and to be pierced through 
with a greater hole ST, of such a magnitude that 
a stream which will exactly fill the lower hole EF 
may pass through it ; the diameter of which hole 
will therefore be to the diameter of the lower hole as 25 to 21, nearly. By 
this means the water will run perpendicularly out at the lower hole ; and 
the quantity of the water running out will be, according to the magnitude 




334 THE MATHEMATICAL PRINCIPLES [BOOK 11 

of this last hole, the same, very nearly, which the solution of the Problem 
requires. The space included between the two planes and the falling stream 
may be considered as the bottom of the vessel. But, to make the solution 
more simple and mathematical, it is better to take the lower plane alone 
for the bottom of the vessel, and to suppose that the water which flowed 
through the ice as through a funnel, and ran out of the vessel through the 
hole EF made in the lower plane, preserves its motion continually, and that 
the ice continues at rest. Therefore in what follows let ST be the diame 
ter of a circular hole described from the centre Z, and let the stream run 
out of the vessel through that hole, when the water in the vessel is all 
fluid. And let EP be the diameter of the hole, which the stream, in fall 
ing through, exactly fills up, whether the water runs out of the vessel by 
that upper hole ST, or flows through the middle of the ice in the vessel, 
as through a funnel. And let the diameter of the upper hole ST be to the 
diameter of the lower EF as about 25 to 21, and let the perpendicular di& 
tance between the planes of the holes be equal to the diameter of the lesser 
hole EF. Then the velocity of the water downwards, in running out of 
the vessel through the hole ST, will be in that hole the same that a body 
may acquire by falling from half the height IZ ; and the velocity of both 
the falling streams will be in the hole EF, the same which a body would 
acquire by falling from the Avhole height IG. 

CASE 2. If the hole EF be not in the middle of the bottom of the ves 
sel, but in some other part thereof, the water will still run out with the 
same velocity as before, if the magnitude of the hole be the same. For 
though an heavy body takes a longer time in descending to the same depth, 
by an oblique line, than by a perpendicular line, yet in both cases it acquires 
in its descent the same velocity ; as Galileo has demonstrated. 

CASE 3. The velocity of the water is the same when it runs out through 
a hole in the side of the vessel. For if the hole be small, so that the in 
terval between the superficies AB and KL may vanish ns to sense, and the 
stream of water horizontally issuing out may form a parabolic figure; from 
the latus rectum of this parabola may be collected, that the velocity of the 
effluent water is that which a body may acquire by falling the height IG 
or HG of the stagnant water in the vessel. For, by making an experi 
ment, I found that if the height of the stagnant water above the hole were 
20 inches, and the height of the hole above a plane parallel to the horizon 
were also 20 inches, a stream of water springing out from thence w r ould 
fall upon the plane, at the distance of 37 inches, very nearly, from a per 
pendicular let fall upon that plane from the hole. For without resistance 
the stream would have fallen upon the plane at the distance of 40 inches, 
the latus rectum of the parabolic stream being 80 inches. 

CASE 4. If the effluent water tend upward, it will still issue forth with 
the same velocity. For the small stream of water springing upward, as- 



SEC. V11.J 



OF NATURAL PHILOSOPHY. 



335 



cends with a perpendicular motion to GH or GI, the height of the stagnant 
water in the vessel ; excepting in so far as its ascent is hindered a little by 
the resistance of the air : and therefore it springs out with the same ve 
locity that it would acquire in falling from that height. Every particle of 
the stagnant water is equally pressed on all sides (by Prop. XIX., Book II), 
and, yielding to the pressure, tends always with an equal force, whether it 
descends through the hole in the bottom of the vessel, or gushes out in an 
horizontal direction through a hole in the side, or passes into a canal, and 
springs up from thence through a little hole made in the upper part of the 
canal. And it may not only be collected from reasoning, but is manifest 
also from the well-known experiments just mentioned, that the velocity 
with which the water runs out is the very same that is assigned in this 
Proposition. 

CASE 5. The velocity of the effluent water is the same, whether the 
figure of the hole be circular, or square, or triangular, or any other figure- 
equal to the circular ; for the velocity of the effluent water does not depend 
upon the figure of the hole, but arises from its depth below the plane 
KL. 

CASE 6. If the lower part of the vessel ABDC 
B be immersed into stagnant water, and the height 
of the stagnant water above the bottom of the ves 
sel be GR, the velocity with which the water that 
is in the vessel will run out at the hole EF into 
the stagnant water will be the same which the 
water would acquire by falling from the height 
IR ; for the weight of all the water in the vessel 
that is below the superficies of the stagnant water 
will be sustained in equilibrio by the weight of the stagnant water, and 
therefore does riot at all accelerate the motion of the descending water in 
the vessel. This case will also appear by experiments, measuring the times 
in which the water will run out. 

COR. 1. Hence if CA the depth of the water be produced to K, so that 
AK may be to CK in the duplicate ratio of the area of a hole made in any 
part of the bottom to the area of the circle AB, the velocity of the effluent 
water will be equal to the velocity which the water would acquire by falling 
from the height KC. 

COR. 2. And the force with which the whole motion of the effluent watei 
may be generated is equal to the weight of a cylindric column of water r 
whose base is the hole EF, and its altitude 2GI or 2CK. For the effluent 
water, in the time it becomes equal to this column, may acquire, by falling 
by its own weight from the height GI, a velocity equal to that with which 
it runs out. 

COR. 3. The weigb t of all the water in the vessel ABDC is to that part 




\ 

336 THE MATHEMATICAL PRINCIPLES [BOOK II 

of the weight which is employed in forcing out the water as the sum of 
the circles AB and EF to twice the circle EF. For let IO be a mean pro 
portional between IH and IG, and the water running out at the hole EF 
will, in the time that a drop falling from I would describe the altitude IG, 
become equal to a cylinder whose base is the circle EF and its altitude 
2IG ; that is, to a cylinder whose base is the circle AB, and whose altitude 
is 2IO. For the circle EF is to the circle AB in the subduplicate ratio cf 
the altitude IH to the altitude IG ; that is, in the simple ratio of the mean 
proportional IO to the altitude IG. Moreover, in the time that a drop 
falling from I can describe the altitude IH, the water that runs out will 
hare become equal to a cylinder whose base is the circle AB, and its alti 
tude 2IH ; and in the time that a drop falling from I through H to G de 
scribes HG, the difference of the altitudes, the effluent water, that is, the 
water contained within the solid ABNFEM, will be equal to the difference 
of the cylinders, that is, to a cylinder whose base is AB, and its altitude 
2HO. And therefore all the water contained in the vessel ABDC is to the 
whole falling water contained in the said solid ABNFEM as HG to2HO, 
that is, as HO + OG to 2HO, or IH + K ) to 2IH. But the weight of all 
the water in the solid ABNFEM is employed in forcing out the water ; 
and therefore the weight of all the water in the vessel is to that part of 
the weight that is employed in forcing out the water as IH + IO to 2IH, 
and therefore as the sum of the circles EF and AB to twice the circle 
EF. 

COR. 4. And hence the weight of all the water in the vessel ABDC is 
to the other part of the weight which is sustained by the bottom of the 
vessel as the sum of the circles AB and EF to the difference of the same 
circles. 

COR. 5. And that part of the weight which the bottom of the vessel sus 
tains is to the other part of the weight employed in forcing out the water 
as the difference of the circles AB and EF to twice the lesser circle EF, or 
as the area of the bottom to twice the hole. 

COR. 6. That part of the weight which presses upon the bottom is to 
the whole weight of the water perpendicularly incumbent thereon as the 
circle AB to the sum of the circles AB and EF, or as the circle AB to thf 
excess of twice the circle AB above the area of the bottom. For that part 
of the weight which presses upon the bottom is to the weight of the whole 
water in the vessel as the difference of the circles AB and EF to the sum 
of the same circles (by Cor. 4) ; and the weight of the whole water in the 
vessel is to the weight of the whole water perpendicularly incumbent on 
the bottom as the circle AB to the difference of the circles AB and EF. 
Therefore, ex ce,quo perturbate, that part of the weight which presses upon 
the bottom is to the weight of the whole water perpendicularly incumbent 



OF NATURAL PHILOSOPHY. 



337 









H 


r 


\ 






\ 


/ 


V 




,* 


\ 


/ 


M 








/ 

/N 










1 










1 

: 


C I 


i ] 


n 


3~ 


Q F I 



thereon as the circle AB to the sum of the circles AB and EF. or the ex 
cess of twice the circle AB above the bottom. 

COR. 7. If in the middle of the hole EF there be placed the little circle 
PQ described about the centre G, and parallel to the horizon, the weight 
of water which that little circle sustains is greater than the weight of a 
third part of a cylinder of water whose base is that little circle and its 
height GH. For let ABNFEM be the cataract or column of falling water 
whose axis is GH, as above, and let all the wa- K ^ 

ter, whose fluidity is not requisite for the ready 
and quick descent of the water, be supposed to 
be congealed, as well round about the cataract, 
as above the little circle. And let PHQ be the 
column of water congealed above the little cir 
cle, whose vertex is H, and its altitude GH. 
And suppose this cataract to fall with its whole 
weight downwards, and not in the least to lie 
against or to press PHQ, but to glide freely by 
it without any friction, unless, perhaps, just at 
the very vertex of the ice, where the cataract at the beginning of its fall 
may tend to a concave figure. And as the congealed water AMEC, BNFD, 
lying round the cataract, is convex in its internal superficies AME, BNF, 
towards the falling cataract, so this column PHQ will be convex towards 
the cataract also, and will therefore be greater than a cone whose base is 
that little circle PQ and its altitude GH; that is, greater than a third 
part of a cylinder described with the same base and altitude. Now that 
little circle sustains the weight of this column, that is, a weight greater 
than the weight of the cone, or a third part of the cylinder. 

COR. 8. The weight of water which the circle PQ ; when very small, sus 
tains, seems to be less than the weight of two thirds of a cylinder of water 
whose base is that little circle, and its altitude HG. For, things standing 
as above supposed, imagine the half of a spheroid described whose base id 
that little circle, and its semi-axis or altitude HG. This figure will be 
equal to two thirds of that cylinder, and will comprehend within it the 
column of congealed water PHQ, the weight of which is sustained by that 
little circle. For though the motion of the water tends directly down 
wards, the external superficies of that column must yet meet the base PQ 
in an angle somewhat acute, because the water in its fall is perpetually ac 
celerated, and by reason of that acceleration become narrower. Therefore, 
oince that angle is less than a right one, this column in the lower parts 
thereof will lie within the hemi-spheroid. In the upper parts also it will be 
acute or pointed; because to make it otherwise, the horizontal motion of 
the water must be at the vertex infinitely more swift than its motion to 
wards the horizon. And the less this circle PQ is, the more acute will 

22 



338 THE MATHEMATICAL PRINCIPLES [BOOK II 

the vertex of this column be ; and the circle being diminished in infinitn/n 
the angle PHQ will be diminished in infinitum, and therefore the co 
lumn will lie within the hemi-spheroid. Therefore that column is less than 
that hemi-spheroid, or than two-third parts of the cylinder whose base is 
that little circle, and its altitude GH. Now the little circle sustains a 
force of water equal to the weight of this column, the weight of the ambient 
water being employed in causing its efflux out at the hole. 

COR. 9. The weight of water which the little circle PQ sustains, when 
it is very small, is very nearly equal to the weight of a cylinder of water 
whose base is that little circle, and its altitude |GH for this weight is an 
arithmetical mean between the weights of the cone and the hemi-spheroid 
above mentioned. But if that little circle be not very small, but on the 
contrary increased till it be equal to the hole EF, it will sustain the weight 
of all the water lying perpendicularly above it, that is, the weight of a 
cylinder of water whose base is that little circle, and its altitude GH. 

COR. 10. Arid (as far as I can judge) the weight which this little circle 
sustains is always to the weight of a cylinder of water whose base is that 
little circle, and its altitude iGH, as EF 2 to EF 2 |PQ 2 , or as the cir 
cle EF to the excess of this circle above half the little circle PQ,, very 
nearly. 

LEMMA IV. 

If a cylinder move uniformly forward in. the direction of its length, the 
resistance made thereto is not at all changed by augmenting or di 
minishing- that length ; and is therefore the same with the resistance 
of a circle, described with the same diameter, and moving forward 
with the same velocity in the direction, of a right line perpendicular to 
its plane. 

For the sides are not at all opposed to the motion ; and a cylinder be 
comes a circle when its length is diminished in infinitum. 

PROPOSITION XXXVII. THEOREM XXIX. 

If a cylinder move uninformly forward in a compressed, infinite, arid 
non-elastic finid, in the direction of its length, the resistance arising 
from the magnitude of its transverse section is to the force by which 
its whole motion may be destroyed or generated, in the time that it 
moves four times its length, as the density of the medium to the den 
sity of the cylinder, nearly. 

For let the vessel ABDC touch the surface of stagnant water with its 
bottom CD, and let the water run out of this vessel into the stagnant wa 
ter through the cylindric canal EFTS perpendicular co the horizon ; and 
let the little circle PQ, be placed parallel to the horizon any where in the 



SEC. VII.] OF NATURAL PHILOSOPHY. 339 

middle of the canal ; and produce CA to K, so K I JL 

f 



that AK may be to CK in the duplicate of the -^ jg "" 



e 



ratio, which the excess of the orifice of the canal 
EF above the little circle PQ bears to the cir 
cle AB. Then it is manifest (by Case 5, Case 
6, and Cor. 1, Prop. XXXVI) that the velocity 
of the water passing through the annular space 
between the little circle and the sides of the ves 
sel will be the very same which the water would 
acquire by falling, and in its fall describing the 
altitude KG or IG. 

And (by Cor. 10, Prop. XXXVI) if the breadth of the vessel be infinite, 
so that the lineola HI may vanish, arid the altitudes IG, HG become equal ; 
the force of the water that flows down and presses upon the circle will be 
to the weight of a cylinder whose base is that little circle, and the altitude 
iIG, as EF 2 to EF 2 |PQ 2 , very nearly. For the force of the water 
flowing downward uniformly through the whole canal will be the same 
upon the little circle PQ. in whatsoever part of the canal it be placed. 

I ,et now the orifices of the canal EF, ST be closed, and let the littk 
circle ascend in the fluid compressed on every side, and by its ascent let it 
oblige the water that lies above it to descend through the annular space 
between the little circle and the sides of the canal. Then will the velocity 
of the ascending little circle be to the velocity of the descending water as 
the difference of the circles EF and PQ, is to the circle PQ; and the ve 
locity of the ascending little circle will be to the sum of the velocities, that 
is, to the relative velocity of the descending water with which it passes by 
the little circle in its ascent, as the difference of the circles EF and PQ to 
the circle EF, or as EF* PQ 2 to EF 2 . Let that relative velocity be 
equal to the velocity with v/hich it was shewn above that the water would 
pass through the annular space, if the circle were to remain unmoved, that 
is, to the velocity which the water would acquire by falling, and in its fall 
describing the altitude IG ; and the force of the water upon the ascending- 
circle will be the same as before (by Cor. 5, of the Laws of Motion) ; that 
is, the resistance of the ascending little circle will be to the weight of a 
cylinder of water whose base is that little circle, and its altitude iIG, as 
EF 2 to EF 2 iPQ 2 , nearly. But the velocity of the little circle will 
be to the velocity which the water acquires by falling, and in its fall de 
scribing the altitude [G, as EF 2 PQ 2 to EF 2 . 

Let the breadth of the canal be increased in wfinitum ; and the ratios 
between EF 2 PQ 2 and EF 2 , and between EF 2 and EF 2 iPQ 2 . 
will become at last ratios of equality. And therefore the velocity of the 
little circle w r ill now be the same which the water would acquire in falling, 
and in its fall describing the altitude IG: and the resistance will become 



340 THE MATHEMATICAL PRINCIPJ ES [BOOK IT. 

equal to the weight of a cylinder whose base is that little circle, and its 
altitude half the altitude IG, from which the cylinder must fall to acquire 
the velocity of the ascending circle ; and with this velocity the cylinder in 
the time of its fall will describe four times its length. But the resistance 
of the cylinder moving forward with this velocity in the direction of its 
length is the same with the resistance of the little circle (by Lem. IV), and 
is therefore nearly equal to the force by which its motion may be generated 
while it describes four times its length. 

If the length of the cylinder be augmented or diminished, its motion, 
and the time in which it describes four times its length, will be augmented 

O t & 

or diminished in the same ratio, and therefore the force by which the mo 
tion so increased or diminished, may be destroyed or generated, will con 
tinue the same ; because the time is increased or diminished in the same 
proportion ; and therefore that force remains still equal to the resistance 
of the cylinder, because (by Lem. IV) that resistance will also remain the 
same. 

If the density of the cylinder be augmented or diminished, its motion, 
and the force by which its motion may be generated or destroyed in the 
same time, will be augmented or diminished in the same ratio. Therefore 
the resistance of any cylinder whatsoever will be to the force by which its 
whole motion may be generated or destroyed, in the time during which it 
moves four times its length, as the density of the medium to the density of 
the cylinder- nearly. Q..E.D. 

A fluid must be compressed to become continued; it must be continued 
and non-elastic, that all the pressure arising from its compression may be 
propagated in an instant ; and so, acting equally upon all parts of the body 
moved, may produce no change of the resistance. The pressure arising 
from the motion of the body is spent in generating a motion in the parts 
of the fluid, and this creates the resistance. But the pressure arising from 
the compression of the fluid, be it ever so forcible, if it be propagated in an 
instant, generates no motion in the parts of a continued fluid, produces no 
change at all of motion therein ; and therefore neither augments nor les 
sens the resistance. This is certain, that the action of the fluid arising 
from the compression cannot be stronger on the hinder parts of the body 
moved than on its fore parts, and therefore cannot lessen the resistance de 
scribed in this proposition. And if its propagation be infinitely swifter 
than the motion of the body pressed, it will not be stronger on the fore 
parts than on the hinder parts. But that action will be infinitely 
swifter, and propagated in an instant, if the fluid be continued and non- 
elastic. 

COR. 1. The resistances, made to cylinders going uniformly forward in 
the direction of their lengths through continued infinite mediums are in a 



com 



A 



Hi 



E 



SEC. VII.] OF NATURAL PHILOSOPHY- 341 

ratio compounded of the duplicate ratio of the velocities and the duplicate 
ratio of the diameters, and the ratio of the density of the mediums. 

COR. 2. If the breadth of the canal be not infinitely increased but the 
cylinder go forward in the direction of its length through an included 
quiescent medium, its axis all the while coinciding with the axis of the 
canal, its resistance will be to the force by which its whole motion, in the 
time in which it describes four times its length, K ............. I... ........ L 

may be generated or destroyed, in a ratio 
pounded of the ratio of EF 2 to EF 2 i 
once, and the ratio of EF 2 to EF 2 PQ, 2 
twice, and the ratio of the density of the medium 
to the density of the cylinder. 

COR. 3. The same thing supposed, and that a 
length L is to the quadruple of the length of 
the cylinder in a ratio compounded of the ratio 
EF 2 -- iPQ 2 to EF 2 once, and the ratio of 
EF 2 PQ, 2 to EF 2 twice; the resistance of 
the cylinder will be to the force by which its whole motion, in the time 
during which it describes the length L, may be destroyed or generated, as 
the density of the medium to the density of the cylinder. 

SCHOLIUM. 

In this proposition we have investigated that resistance alone which 
arises from the magnitude of the transverse section of the cylinder, neg 
lecting that part of the same which may arise from the obliquity of the 
motions. For as, in Case 1, of Prop. XXXVL, the obliquity of the mo 
tions with which the parts of the water in the vessel converged on every 
side to the hole EF hindered the efflux of the water through the hole, so, 
in this Proposition, the obliquity of the motions, with which the parts of 
the water, pressed by the antecedent extremity of the cylinder, yield to the 
pressure, and diverge on all sides, retards their passage through the places 
that lie round that antecedent extremity, toward the hinder parts of the 
cylinder, and causes the fluid to be moved to a greater distance; which in 
creases the resistance, and that in the same ratio almost in which it dimin 
ished the efflux of the water out of the vessel, that is, in the duplicate ratio 
of 25 to 21, nearly. And as, in Case 1, of that Proposition, we made the 
parts of the water pass through the hole EF perpendicularly and in the 
greatest plenty, by supposing all the water in the vessel lying round the 
cataract to be frozen, and that part of the water whose motion was oblique, 
and useless to remain without motion, so in this Proposition, that the 
obliquity of the motions may be taken away, and the parts of the water 
may give the freest passage to the cylinder, by yielding to it witli the most 
direct and quick motion possible, so that only so much resistance may re- 



542 THE MATHEMATICAL PRINCIPLES [BoOK II. 

main as arises from the magnitude of the transverse section, and which is 
incapable of diminution, unless by diminishing the diameter of the cylinder ; 
we must conceive those parts of the fluid whose motions are oblique and 
useless, and produce resistance, to be at rest among themselves at both ex 
tremities of the cylinder, and there to cohere, and be joined to the cylinder. 
Let ABCD be a rectangle, and let 

AE and BE be two parabolic arcs, i 1 

described with the axis AB, and g j^ 

with a latus rectum that is to the .----"" 
space HG, which must be described 
by the cylinder in falling, in order 
to acquire the velocity with which it moves, as HG to ^AB. Let CF and 
DF be two other parabolic arcs described with the axis CD, and a latus 
rectum quadruple of the former; and by the convolution of the figure 
about the axis EF let there be generated a solid, whose middle part ABDC 
is the cylinder we are here speaking of, and whose extreme parts ABE and 
CDF contain the parts of the fluid at rest among themselves, and concreted 
into two hard bodies, adhering to the cylinder at each end like a head and 
tail. Then if this solid EACFDB move in the direction of the length of 
its axis FE toward the parts beyond E, the resistance will be the same 
which we have here determined in this Proposition, nearly ; that is, it will 
have the same ratio to the force with which the whole motion of the cyl 
inder may be destroyed or generated, in the time that it is describing the 
length 4AC with that motion uniformly continued, as the density of the 
fluid has to the density of the cylinder, nearly. And (by Cor. 7, Prop. 
XXXVI) the resistance must be to this force in the ratio of 2 to 3, at the 
least. 

LEMMA V. 

If a cylinder, a sphere, and a spheroid, of equal breadths be placed suc 
cessively in the middle of a cylindric canal, so that their axes may 
coincide with the axis of the canal, these bodies will equally hinder t^e 
passage of the water through the canal. 

For the spaces lying between the sides of the canal, and the cylinder, 
sphere, and spheroid, through which the water passes, are equal ; and the 
water will pass equally through equal spaces. 

This is true, upon the supposition that all the water above the cylinder, 
sphere, or spheroid, whose fluidity is not necessary to make the passage of 
the water the quickest possible, is congealed, as was explained above in Cer 
7, Prop. XXXVI. 



SEC. VII.] OF NATURAL PHILOSOPHY 343 

LEMMA VI. 

The same supposition remaining, the fore- mentioned bodies are equally 

acted OIL by the water Jlowin g through the canal. 
This appears by Lein. V and the third Law. For tht water and the 
bodies act upon each other mutually and equally. 

LEMMA VIL 

If the water be at rest in the canal, and these bodies move with equil ve 
locity and the contrary way through the canal, their resistances will 
be equal among themselves. 
This appears from the last Lemma, for the relative motions remain the 

same among themselves. 

SCHOLIUM. 

The case is the same of all convex and round bodies, whose axes coincide 
with the axis of the canal. Some difference may arise from a greater or 
less friction; but in these Lemmata we suppose the bodies to be perfectly 
smooth, and the medium to be void of all tenacity and friction ; and that 
those parts of the fluid which by their oblique and superfluous motions may 
disturb, hinder, and retard the flux of the water through the canal, are at 
rest amorg themselves ; being fixed like water by frost, and adhering to 
the fore and hinder parts of the bodies in the manner explained in the 
Scholium of the last Proposition : for in what follows we consider the very 
least resistance that round bodies described with the greatest given trans 
verse sections can possibly meet with. 

Bodies swimming upon fluids, when they move straight forward, cause 
the fluid to ascend at their fore parts and subside at their hinder parts, 
especially if they are of an obtuse figure ; and thence they meet with a 
little more resistance than if they were acu*-e at the head and tail. And 
bodies moving in elastic fluids, if they are obtuse behind and before, con 
dense the fluid a little more at their fore parts, and relax the same at theii 
hinder parts ; and therefore meet also with a little more resistance than ii 
they were acute at the head and tail. But in these Lemmas and Proposi 
tions we are not treating of elastic but non-elastic fluids; not of bodies 
floating on the surface of the fluid, but deeply immersed therein. And 
when the resistance of bodies in non-elastic fluids is once known, we may 
then augment this resistance a little in elastic fluids, as our air; and in 
the surfaces of stagnating fluids, as lakes and seas. 

PROPOSITION XXXVIII. THEOREM XXX. 

If a globe move uniformly forward in a compressed, infinite, and no?t 
elastic fluid, its resistance is to the force by which its whole 



544 THE MATHEMATICAL PRINCIPLES [BOOK II 

may be destroyed or generated, in the time that it describes eight third 

parts of its diameter, as the density of the fluid to the density of the 

globe, very nearly. 

For the globe is to its circumscribed cylinder as two to three ; and there 
fore the force which can destroy all the motion of the cylinder, while the 
same cylinder is describing the length of four of its diameters, will destroy 
all the motion of the globe, while the globe is describing two thirds of this 
length, that is, eight third parts of its own diameter. Now the resistance 
of the cylinder is to this force very nearly as the density of the fluid to the 
density of the cylinder or globe (by Prop. XXXVII), and the resistance of 
the globe is equal to the resistance of the cylinder (by Lem. V, VI, and 
VII). Q.E.D. 

COR. I. The resistances of globes in infinite compressed mediums are in 
a ratio compounded of the duplicate ratio of the velocity, and the dupli 
cate ratio of the diameter, and the ratio of the density of the mediums. 

COR. 2. The greatest velocity, with which a globe can descend by its 
comparative weight through a resisting fluid, is the same which it may 
acquire by falling with the same weight, and without any resistance, and 
in its fall describing a space that is, to four third parts of its diameter as 
the density of the globe to the density of the fluid. For the globe in the 
time of its fall, moving with the velocity acquired in falling, will describe 
a space that will be to eight third parts of its diameter as the density of 
the globe to the density of the fluid ; and the force of its weight which 
generates this motion will be to the force that can generate the same mo 
tion, in the time that the globe describes eight third parts of its diameter, 
with the same velocity as the density of the fluid to the density of the 
globe; and therefore (by this Proposition) the force of weight will be equal 
to the force of resistance, and therefore cannot accelerate the globe. 

COR. 3. If there be given both the density of the globe and its velocity 
at the beginning of the motion, and the density of the compressed quiescent 
fluid in which the globe moves, there is given at any time both the velo 
city of the globe and its resistance, and the space described by it (by Cor. 
7, Prop. XXXV). 

COR. 4. A globe moving in a compressed quiescent fluid of the same 
density with itself will lose half its motion before it can describe the length 
of two of its diameters (by the same Cor. 7). 

PROPOSITION XXXIX. THEOREM XXXI. 

If a S lobe move uniformly forward through a fluid inclosed and com 
pressed in a cylindric canal, its resistance is to the force by which its 
whole motion may be generated or destroyed, in the time in which it 
describes eight third parts of its dia?neter t in a ratio compounded of 



EC. VII.] OF NATURAL PHILOSOPHY. 345 

the ratio of the orifice of the canal to the excess of that orifice above 
half the greatest circle of the globe ; and the duplicate ratio of the 
orifice of the canal, to the excess of that orifice above the greatest circle 
of the globe ; and t/ie ratio of the density of the fluid to the density of 
the globe, nearly. 

This appears by Cor. 2, Prop. XXXVII, and the demonstration pro 
ceeds in the same manner as in the foregoing Proposition. 

SCHOLIUM. 

In the last two Propositions we suppose (as was done before in Lem. V) 
that all the water which precedes the globe, and whose fluidity increases 
the resistance of the same, is congealed. Now if that water becomes fluid, 
it will somewhat increase the resistance. But in these Propositions that 
increase is so small, that it may be neglected, because the convex superfi 
cies of the globe produces the very same effect almost as the congelation 
of the water. 

PROPOSITION XL. PROBLEM IX. 

To find by phenomena the resistance of a globe moving through a per 
fectly fluid compressed medium. 

Let A be the weight of the globe in vacua, B its weight in the resisting 
medium, D the diameter of the globe. F a space which is to f D as the den 
sity of the globe to the density of the medium, that is, as A to A B, G 
the time in which the globe falling with the weight B without resistance 
describes the space P, and H the velocity which the body acquires by that 
fall. Then H will be the greatest velocity with which the globe can pos 
sibly descend with the weight B in the resisting medium, by Cor. 2, Prop 
XXXVIII ; and the resistance which the globe meets with, when descend 
ing with that velocity, will be equal to its weight B ; and the resistance it 
meets with in any other velocity will be to the weight B in the duplicate ra 
tio of that velocity to the greatest velocity H, by Cor. 1, Prop. XXXVIII. 

This is the resistance that arises from the inactivity of the matter of 
the fluid. That resistance which arises from the elasticity, tenacity, and 
friction of its parts, may be thus investigated. 

Let the globe be let fall so that it may descend in the fluid by the weight 
B ; and let P be the time of falling, and let that time be expressed in sec 
onds, if the time G be given in seconds. Find the absolute number N 

2P 

agreeing to the logarithm 0,4342944819 > an d let L be the logarithm of 

N 4- 1 
the number an d *^ e velocity acquired in falling will bf 



THE MATHEMATICAL PRINCIPLES 



[BOOK 11 



N i 2PF 

jj= H, and the height described will be -^ 1 .38629430 IIP + 

4,6051701S6LF. If the fluid be of a sufficient depth, we may neglect the 

2PF 
term 4,6051 70186LF; and r - 1,3362943611F will be the altitude 

described, nearly. These things appear by Prop. IX, Book II, and its Corol 
laries, and are true upon this supposition, that the globe meets with no other 
resistance but that which arises from the inactivity of matter. Now if it 
really meet with any resistance of another kind, the descent will be slower, 
and from the quantity of that retardation will be known the quantity of 
this new resistance. 

That the velocity and descent of a body falling in a fluid might more 
easily be known, I have composed the following table ; the first column of 
which denotes the times of descent ; the second shews the velocities ac 
quired in falling, the greatest velocity being 100000000: the third exhib 
its the spaces described by falling in those times, 2F being the space which 
the body describes in the time G with the greatest velocity ; and the fourth 
gives the spaces described with the greatest velocity in the same times. 

2P 

The numbers in the fourth column are -pn and by subducting the number 

1,3962944 4,60517021,, are found the numbers in the third column ; 
and these numbers must be multiplied by the space F to obtain the spaces 
described in falling. A fifth column is added to all these, containing the 
spaces described in the same times by a body falling in vacno with the 
force of B its comparative weight, 



The Times 
P. 


Velocities of the 
body falling 
in the fluid. 


The spaces de 
sensed in fall 
ing in the fluid. 


The spaces descri 
bed with the 
grea est motion. 


The spaces de 
scribed hy fall 
ing 1 in vacua. 


0,001G 


99999|ii 


O.OOOOOlF 


0,002F 


O.OOOOOlF 


0,01G 


999967 


0,0001F 


0,02F 


O^OOOlF 


0,1G 


9966799 


0,0099834 F 


0.2F 


0,01F 


0^2G 


19737532 


0,039736 IF 


0,4F 


0.04F 


0,3G 


29131261 


o!o86815F 


0.6F 


0^09F 


3 4G 


37994896 


,1559070F 


0.8F 


OJ1 6F 


0,5G 


46211716 


0.240-2290F 


lioF 


0,25F 


0,6G 


53704957 


0^3402706F 


1,2F 


0.36 F 


0,7G 


60436778 


0.4545405F 


1.4F 


0.1 9F 


0,8G 


66403677 


0,581507lF 


1,6F 


0,64F 


0.9G 


71629787 


0.7196609F 


1,SF 


0.8 IF 


1G 


76159416 


0.8675617F 


2F 


IF 


2G 


96402758 


^6500055F 


4F 


4F 


3G 


99505475 


4.6186570F 


6F 


9F 


4G 


99932930 


6.6143765F 


8F 


16F 


5G 


99990920 


8!6137964F 


10F 


25F 


6G 


99998771 


10,6137179F 


12F 


36F 


7G 


99999834 


12.6137073F 


14F 


49F 


8G 


99999980 


14!6137059F 


16F 


64F 


9G 


99999997 


16.6137057F 


18F 


81F 


IOG 


99999999| 


18 ; 6137056F 


20F 


100F 



SEC. VII. | OF NATURAL PHILOSOPHY. 347 

\ 

SCHOLIUM* 

In order to investigate the resistances of lluids from experiments, I pro 
cured a square wooden vessel, whose length and breadth on the inside was 
9 inches English measure, and its depth 9 feet \ ; this I filled with rain 
water: and having provided globes made up of wax, and lead included 
therein, I noted the times of the descents of these globes, the height through 
which they descended being 112 inches. A solid cubic foot of English 
measure contains 76 pounds troy weight of rain water ; and a solid inch 
contains if ounces troy weight, or 253 grains: and a globe of water of 
one inch in diameter contains 132,645 grains in air, or 132,8 grains in 
vacn.o ; and any other globe will be as the excess of its weight in vacuo 
above its weight in water. 

EXPER. 1. A globe whose weight was 156^ grains in air, and 77 grains 
in water, described the whole height of 1 12 inches in 4 seconds. And, upon 
repeating the experiment, the globe spent again the very same time of 4 
seconds in falling. 

The weight of this globe in vacuo is 156^1 grains; and the excess of 
this weight above the weight of the globe in water is 79^ f grains. Hence 
the diameter of the globe appears to be 0,84224 parts of an inch. Then it 
will be, as that excess to the weight of the globe in vacuo, so is the density 
of the water to the density of the globe; and so is f parts of the diameter 
of the globe (viz. 2,24597 inches) to the space 2F, which will be therefore 
4.4256 inches. Now a globe falling in vacuo with its whole weight of 
156^f grains in one second of time will describe 193| inches ; and falling 
in water in the same time with the weight of 77 grains without resistance, 
will describe 95,219 inches ; and in the time G, which is to one second of 
time in the subduplicate ratio of the space P, or of 2,2128 inches to 95,219 
inches, will describe 2,2128 inches, and will acquire the greatest velocity H 
with which it is capable of descending in water. Therefore the time G is 
0",15244. And in this time G, with that greatest velocity H, the globe 
will describe the space 2F, which is 4,4256 inches; and therefore in 4 sec 
onds will describe a space of 1 16,1245 inches. Subduct the space 1, 3862944 F, 
or 3,0676 inches, and there will remain a space of 113,0569 inches, which 
the globe falling through water in a very wide vessel will describe in 4 sec 
onds. But this space, by reason of the narrowness of the wooden vessel 
before mentioned, ought to be diminished in a ratio compounded of the sub- 
duplicate ratio of the orifice of the vessel to the excess of this orifice above 
half a great circle of the globe, and of the simple ratio of the same orifice 
to its excess above a great circle of the globe, that is, in a ratio of 1 to 
0,9914. This done, we have a space of 112,08 inches, which a globe fall 
ing through the water in this wooden vessel in 4 seconds of time ought 
nearly to describe by this theory; but it described 112 inches by the ex 
periment. 



348 THE MATHEMATICAL PRINCIPLES [BOOK II 

EXPER. 2. Three equal globes, whose weights were severally 76^- grains 
in air, and 5 T ^ grains in water, were let fall successively -; and every one 
fell through the water in 15 seconds of time, describing in its fall a height 
of 112 inches. 

By computation, the weight of each globe in vacuo is 76 T 5 2 grains ; the 
excess of this weight above the weight in water is 71 grains J ; the diam 
eter of the globe 0,81296 of an inch; f parts of this diameter 2,1 67S 
inches; the space 2F is 2,3217 inches; the space which a globe of 5 T \ 
grains in weight would describe in one second without resistance, 12,80 
inches, and the time G0",301056. Therefore the globe, with the greatest 
velocity it is capable of receiving from a weight of 5^ grains in its de 
scent through water, will describe in the time 0",3L)1056the space of 2,3217 
inches; and in 15 seconds the space 115,678 inches. Subduct the space 
1,3862944F, or 1,609 inches, and there remains the space 114.069 inches, 
which therefore the falling globe ought to describe in the same time, if the 
vessel were very wide. But because our vessel was narrow, the space ought 
to be diminished by about 0,895 of an inch. And so the space will remain 
113,174 inches, which a globe falling in this vessel ought nearly to de 
scribe in 15 seconds, by the theory. But by the experiment it described 
112 inches. The difference is riot sensible. 

EXPER. 3. Three equal globes, whose weights were severally 121 grains 
in air, and 1 grain in water, were successively let fall ; and they fell 
through the water in the times 46", 47", and 50", describing a height oi 
112 inches. 

By the theory, these globes ought to have fallen in about 40". Now 
whether their falling more slowly were occasioned from hence, that in slow 
motions the resistance arising from the force of inactivity does really bear 
a less proportion to the resistance arising from other causes ; or whether 
it is to be attributed to little bubbles that might chance to stick to the 
globes, or to the rarefaction of the wax by the warmth of the weather, or 
of the hand that let them fall ; or, lastly, whether it proceeded from some 
insensible errors in weighing the globes in the water, I am not certain. 
Therefore the weight of the globe in water should be of several grains, that 
the experiment may be certain, and to be depended on. 

EXPER. 4. I began the foregoing experiments to investigate the resistan 
ces of fluids, before I was acquainted with the theory laid down in the 
Propositions immediately preceding. Afterward, in order to examine the 
theory after it was discovered, I procured a wooden vessel, whose breadth 
on the inside was Sf inches, and its depth ] 5 feet and -i. Then I made 
four globes of wax, with lead included, each of which weighed 139 1 grains 
in air, and 7\ grains in water. These I let fall, measuring the times of their 
falling in the water with a pendulum oscillating to half seconds. The 
globes were cold, and had remained so some time, both when they were 



SEC. V1L] OF NATUKAL PHILOSOPHY. 3-1 *J 

.reighed and when they were let fall ; because warmth rarefies the wax. and 
by rarefying it diminishes the weight of the globe in the water ; and wax, 
when rarefied, is not instantly reduced by cold to its former density. Be 
fore they were let fall, they were totally immersed under water, lest, by the 
weight of any part of them that might chance to be above the water, their 
descent should be accelerated in its beginning. Then, when after their 
immersion they were perfectly at rest, they were let go with the greatest 
care, that they might not receive any impulse from the hand that let them 
down. And they fell successively in the times of 47 J, 48^, 50, and 51 os 
cillations, describing a height of 15 feet and 2 inches. But the weather 
was now a little colder than when the globes were weighed, and therefore 1 
repeated the experiment another day ; and then the globes fell in the times 
of 49, 49i, 50. and 53; and at a third trial in the times of 49, 50, 51. 
and 53 oscillations. And by making the experiment several times over, I 
found that the globes fell mostly in the times of 49| and 50 oscillations. 
When they fell slower, I suspect them to have been retarded by striking 
against the sides of the vessel. 

Now, computing from the theory, the weight of the globe in vacno is 
139| grains; the excess of this weight above the weight of the globe in 
water 132|i grains ; the diameter of the globe 0,99868 of an inch : |- parts 
of the diameter 2,66315 inches; the space 2F 2,8066 inches; the space 
which a globe weighing 7| grains falling without resistance describes in a 
second of time 9,88164 inches; and the time G0",376843 Therefore the 
globe with the greatest velocity with which it is capable of descending 
through the water by the force of a weight of 7} grains, will in the time 
0",376843 describe a space of 2,8066 inches, and in one second of time a 
space of 7,44766 inches, and in the time 25", or in 50 oscillations, the space 
186,1915 inches. Subduct the space 1,386294F, or 1,9454 inches, and 
there will remain the space 184,2461 inches which the globe will describe 
in that time in a very wide vessel. Because our vessel was narrow, let this 
space be diminished in a ratio compounded of the subduplicate ratio of the 
orifice of the vessel to the excess of this orifice above half a great circle of 
the globe, and of the simple ratio of the same orifice to its excess above a 
great circle of the globe ; and we shall have the space of 181,86 inches, 
which the globe ought by the theory to describe in this vessel in the time 
of 50 oscillations, nearly. But it described the space of 182 inches, by 
experiment, in 49^ or 50 oscillations. 

EXPER. 5. Pour globes weighing 154| grains in air, and 21 1 grains in 
water, being let fall several times, fell in the times of 28^, 29, 29 , and 30, 
and sometimes of 31, 32, and 33 oscillations, describing a height of 15 feet 
and 2 inches. 

They ought by the theory to have fallen in the time of 29 oscillations, 
nearly. 



350 THE MATHEMATICAL PRINCIPLES | BOOK IL 

EXPER. 6. Five globes, weighing 212f grains in air, and 79^ in water, 
being several times let fall, fell in the times of 15, 15^, 16, 17, and 18 os 
cillations, describing a height of 15 feet and 2 inches. 

By the theory they ought to have fallen in the time cf 15 oscillations, 
nearly. 

EXPER. 7. Four globes, weighing 293 f grains in air, and 35| grains in 
water, being let fall several times, fell in the times of 29 30, 301 31, 32, 
and 33 oscillations, describing a height of 15 feet and 1 inch and . 

By the theory they ought to have fallen in the time of 28 oscillations, 
nearly. 

In searching for the cause that occasioned these globes of the same weight 
and magnitude to fall, some swifter and some slower, I hit upon this ; that 
the globes, when they were first let go and began to fall, oscillated about 
their centres; that side which chanced to be the heavier descending first, 
and producing an oscillating motion. Now by oscillating thus, the globe 
communicates a greater motion to the water than if it descended without 
any oscillations ; and by this communication loses part of its own motion 
with which it should descend ; and therefore as this oscillation is greater 
or less, it will be more or less retarded. Besides, the globe always recedes 
from that side of itself which is descending in the oscillation, and by so 
receding comes nearer to the sides of the vessel, so as even to strike against 
them sometimes. And the heavier the globes are, the stronger this oscil 
lation is ; and the greater they are, the more is the water agitated by it. 
Therefore to diminish this oscillation of the globes, 1 made new ones of 
lead and wax, sticking the lead in one side of the globe very near its sur 
face; and I. let fall the globe in such a manner, that, as near as possible, 
the heavier side might be lowest at the beginning of the descent. By this 
means the oscillations became much less than before, and the times in which 
the globes fell were not so unequal: as in the following experiments. 

EXPER. 8. Four globes weighing 139 grains in air, and 6| in water, 
were let fall several times, and fell mostly in the time of 51 oscillations, 
never in more than 52, or in fewer than 50, describing a height of 182 
inches. 

By the theory they ought to fall in about the time of 52 oscillations 

EXPER. 9. Four globes weighing 273^ grains in air, and 140 in water, 
being several times let fall, fell in never fewer than 12, and never more 
than 13 oscillations, describing a height of 182 inches. 

These globes by the theory ought to have fallen in the time of 1 1 } os 
cillations, nearly. 

EXPER. 10. Four globes, weighing 384 grains in air, and 119^ in water, 
oeing let fall several times, fell in the times of 17f 18, 18^, and 19 oscilla 
tions, descril ing a height of 181| inches. And when they fell in the time 



SEC. VI1.J OF NATURAL PHILOSOPHY. 351 

of 19 oscillations, I sometimes heard them hit against the sides of tl.e ves 
sel before they reached the bottom. 

By the theory they ought to have fallen in the time of 15f oscillations, 
nearly. 

EXPER. 11. Three equal globes, weighing 48 grains in the air, and 3|| 
in water, being several times let fall, fell in the times of 43J, 44, 44 1, 45, 
and 46 oscillations, and mostly in 44 and 45. describing a height of 182* 
inches, nearly. 

By the theory they ought to have fallen in the time of 46 oscillations 
and f, nearly. 

EXPER. 12. Three equal globes, weighing 141 grains in air, and 4| in 
water, being let fall several times, fell in the times of 61, 62, 63, 64, and 
65 oscillations, describing a space of 182 inches. 

And by the theory they ought to have fallen in 641 oscillations 
nearly. 

From these experiments it is manifest, that when the globes fell slowly, 
as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments; 
the times of falling are rightly exhibited by the theory but when the 
globes fell more swiftly, as in the sixth, ninth, and tenth experiments, the 
resistance was somewhat greater than in the duplicate ratio of the velocity. 
For the globes in falling oscillate a little : and this oscillation, in those 
globes that are light and fall slowly, soon ceases by the weakness of the 
motion ; but in greater and heavier globes, the motion being strong, it con 
tinues longer, and is not to be checked by the ambient water till after sev 
eral oscillations Besides, the more swiftly the globes move, the less are 
they pressed by the fluid at their hinder parts; and if the velocity be. per 
petually increased, they will at last leave an empty space behind them, 
unless the compression of the fluid be increased at the same time. For the 
compression of the fluid ought to be increased (by Prop. XXXII and 
XXXIII) in the duplicate ratio of the velocity, in order to preserve the re 
sistance in the same duplicate ratio. But because this is not done, the 
globes that move swiftly are not so much pressed at their hinder parts as 
the others; and by the defect of this pressure it comes to pass that their 
resistance is a little greater than in a duplicate ratio of their velocity. 

So that the theory agrees with the phenomena of bodies falling in water 
It remains that we examine the phenomena of bodies falling in air. 

EXPER. 13. From the top of St. Paul s Church in London, in Juiib 
1710, there w.ere let fall together two glass globes, one full of quicksilver, 
the other of air; and in their fall they described a height of 220 English 
feet. A wooden table was suspended upon iron hinges on one sidi> and the 
other side of the same was supported by a wooden pin. The twn globes 
lying upon this table were let fall together by pulling out the pin bj 
means of an iron wire reaching from thence quite down to the ground ; s< 



352 



THE MATHEMATICAL PRINCIPLES 



[BOOK II, 



that, the pin being removed, the table, which had then no support but the 
iron hinges, fell downward, and turning round upon the hinges, gave leave 
to the globes to drop off from it. At the same instant, with the same pull 
of the iron wire that took out the pin, a pendulum oscillating to seconds 
was let go, and began to oscillate. The diameters and weights of the 
globes, and their times of falling, are exhibited in the following table. 



ike globes filled with mere 
Weights. DiaiTV ters 


ury. 

Times i, 
falling. 


T/ic globes full of < 

Weighs Diameters. 


lir. 
Times in 
falling 


908 grains 
983 
866 
747 
808 
784 


0.8 of an inch 

0,8 
0.8 
0.75 
0.75 
0.75 


4" 
4 
4 
4 + 
4 
4 + 


510 grains 
642 
599 
515 
483 
641 


5,1 inches 
5,2 
5.1 

s;o 

5,0 

3 ; 2 


? Ht-O 

oo oo oo oo oo oo 



But the times observed must be corrected ; for the globes of mercury (by 
Galileo s theory), in 4 seconds of time, will describe 257 English feet, and 
220 feet in only 3"42 ". So that the wooden table, when the pin was taken 
out, did not turn upon its hinges so quickly as it ought to have done; and 
the slowness of that revolution hindered the descent of the globes at the 
beginning. For the globes lay about the middle of the table, and indeed 
were rather nearer to the axis upon which it turned than to the pin. And 
hence the times of falling were prolonged about 18 "; and therefore ought 
to be corrected by subducting that excess, especially in the larger globes, 
which, by reason of the largeness of their diameters, lay longer upon the 
revolving table than the others. This being done, the times in which the 
six larger globes fell will come forth 8" 12 ", 7" 42% 7" 42 ", 7" 57 ", 8" 12 " 
and 7" 42 ". 

Therefore the fifth in order among the globes that were full of air being 
5 inches in diameter, and 483 grains in weight, fell in 8" 12 ", describing a 
space of 220 feet. The weight of a bulk of water equal to this globe is 
1 6600 grains; and the weight of an equal bulk of air is l ||f- grains, or I9 r 3 o 
grains ; and therefore the weight of the globe in vacuo is 502 T 3 ?r grains; 
and this weight is to the weight of a bulk of air equal to the globe as 
502 T ; v to 19 T 3 o- ; and so is 2P to | of the diameter of the globe, that is, to 
13i inches. Whence 2F becomes 28 feet 11 inches. A globe, falling in 
vacuo with its whole weight of 502 T 3 grains, will in one second of time 
describe 193| inches as above ; and with the weight of 483 grains will de 
scribe 185,905 inches; and with that weight 483 grains in vacuo will de 
scribe the space F, or 14 feet 5i inches, in the time of 57 " 58"", and ac 
quire the greatest velocity it is capable of descending with in the air. 
With this velocity the globe in 8" 12 " of time will describe 245 feet and 
5i inches. Subduct 1,3863F, or 20 feet and | an inch, and there remain 
225 feet 5 inches. This space, therefore, the falling globe ought by the 



SEC. VIIJ 



OF NATURAL PHILOSOPHY 



theory to describe in 8" 12 ". But* by the experiment it descrioed a space 
of 220 feet. The difference is insensible. 

By like calculations applied to the other globes full of air, I composed 



the following table. 



The weitf it< 
of the glol e 


Th^ diame 
ter > 


n,e times ul 
all-ng from 
a h. U lit o 
2-2( lent 

8 1* 

7 42 


T> e spacf< which they 
wuiill de*e i it>e by the 
heory 


The excesses. 


510 iiraini 5 
642^ 


5.1 inches 
5,2 


226 feet 11 inch. 
230 9 


6 feet 11 _nch. 
10 9 


599 
515 


5,1 
5 


7 42 |227 10 

7 57 224 5 


7 
4 5 


483 


5 


8 12 


225 5 


5 5 


641 


5,2 


7 42 


|230 7 


10 7 



EXPER. 14. Anno 1719, in the month of July, Dr. Desaguliers made 
some experiments of this kind again, by forming hogs bladders into spheri 
cal orbs ; which was done by means of a concave wooden sphere, which the 
bladders, being wetted well first, were put into. After that being blown 
full of air. they were obliged to fill up the spherical cavity that contained 
them ; and then, when dry, were taken out. These were let fall from the 
lantern on the top of the cupola of the same church, namely, from a height 
of 272 feet ; and at the same moment of time there was let fall a leaden 
globe, whose weight was about 2 pounds troy weight. And in the mean 
time some persons standing in the upper part of the church where the 
globes were let fall observed the whole times of falling ; and others stand 
ing on the ground observed the differences of the times between the fall 
of the leaden weight and the fall of the bladder. The times were measured 
by pendulums oscillating to half seconds. And one of those that stood 
upon the ground had a machine vibrating four times in one second ; and 
another had another machine accurately made with a pendulum vibrating 
four times in a second also. One of those also who stood at the top of the 
church had a like machine ; and these instruments were so contrived, that 
their motions could be stopped or renewed at pleasure. Now the leaden 
globe fell in about four seconds and i of time; and from the addition of 
this time to the difference of time above spoken of, was collected the \Vhole 
time in which the bladder was falling. The times which the five bladders 
spent in falling, after the leaden globe had reached the ground, were, tn*e 
first time, 14", 12f, 14f , 17 f, and 16J-" ; and the second time, 14i", 14}", 
14", 19", and 16 J". Add to these 4", the time in which the leaden globe 
was falling, and the whole times in which the five bladders fell were, the 
first fane, 19* 17", 18J", 22", and 21}"; and the second time, 18f, 18i", 
ISj", 23{", and 21". The times observed at the top of the church were, 
the first time, 19 f", 17f , 18f, 22f , and 21f"; and the second time, 19", 
ISf", ISf, 24". and 211". But the bladders did not always fall directly 
down, but sometimes fluttered a little in the air, and waved to and fro, aa 



354 



THE MATHEMATICAL PRINCIPLES 



[BOOK Jl 



they were descending. And by these motions the times of their falling 
were prolonged, and increased by half a second sometimes, and sometimes 
by a whole second. The second and fourth bladder fell most directly the 
first time, and the first and third the second time. The fifth bladder was 
wrinkled, and by its wrinkles was a little retarded. I found their diame 
ters by their circumferences measured with a very fine thread wound about 
them twice. In the following table I have compared the experiments with 
the theory ; making the density of air to be to the density of rain-water as 
1 to 860, and computing the spaces which by the theory the globes ought 
to describe in falling. 



The weight 
of tlie bla !- 


The diameters 


Che times ol 
falling from 
a height oi 


The spaces winch by 
the theory ought to 
have been described 


The difference be 
tween the theory 
and the experi 







272 t, t-t 


in those times 


ments 


128 grains 


5.28 inches 


19" 


271 feet 11 in. 


it. 1 in. 


156 


5.19 


17 


272 


-f o 04 


137* 


5.3 


18 


272 7 


-f 7 


97d 


5.26 


22 


277 4 


+ 5 4 


99& 


5 


21| |J282 |+ 10 



Our theory, therefore, exhibits rightly, within a very little, all the re 
sistance that globes moving either in air or in water meet with ; which^ap- 
pears to be proportional to the densities of the fluids in globes of equal ve 
locities and magnitudes. 

In the Scholium subjoined to the sixth Section, we shewed, by experi 
ments of pendulums, that the resistances of equal and equally swift globes 
moving in air, water, and quicksilver, are as the densities of the fluids. 
We here prove the same more accurately by experiments of bodies falling 
in air and water. For pendulums at each oscillation excite a motion in 
the fluid always contrary to the motion of the pendulum in its return : and 
the resistance arising from this motion, as also the resistance of the thread 
by which the pendulum is suspended, makes the whole resistance of a pen 
dulum greater than the resistance deduced from the experiments of falling 
bodies. For by the experiments of pendulums described in that Scholium, 
a globe of the same density as water in describing the length of its semi- 
diameter in air would lose the -3^-0 part of its motion. But by the 
theory delivered in this seventh Section, and confirmed by experiments of 
falling bodies, the same globe in describing the same length would lose only 
a part of its motion equal to j-Vir? supposing the density of water to be 
to the density of air as 8 r >0 to 1. Therefore the resistances were found 
greater by the experiments of pendulums (for the reasons just mentioned) 
than by the experiments of falling globes ; and that in the ratio of about 
4 to 3. Bat yet since the resistances of pendulums oscillating in air, wa 
ter, and quicksilver, are alike increased by like causes, the proportion of 
the resistances in these mediums will be rightly enough exhibited by th 



SEC. VII.J OF NATUKAL PHILOSOPHY. 355 

experiments of pendulums, as well as by the experiments of falling bodies. 
And from all this it may be concluded, that the resistances of bodies, moving 
in any fluids whatsoever, though of the most extreme fluidity, are, cceteris 
paribus, as the densities of the fluids. 

These things being thus established, we may now determine what part 
of its motion any globe projected in any fluid whatsoever would nearly lose 
in a given time. Let D be the diameter of the globe, and V its velocity 
at the beginning of its motion, and T the time in which a globe with the 
velocity V can describe in vacua a space that is, to the space |D as the 
density of the globe to the density of the fluid ; and the globe projected 

*V 

in that fluid will, in any other time t lose the part , the part 

1 -p t 

TV 

r remaining ; and will describe a space, which will be to that de 

scribed in the same time in, vacua with the uniform velocity V, as the 

T + t 
logarithm of the number ~ multiplied by the number 2,302585093 is 

to the number 7^, by Cor. 7, Prop. XXXV. In slow motions the resist 
ance may be a little less, because the figure of a globe is more adapted to 
motion than the figure of a cylinder described with the same diameter. In 
swift motions the resistance may be a little greater, because the elasticity 
and compression of the fluid do not increase in the duplicate ratio of the 
velocity. But these little niceties I take no notice of. 

And though air. water, quicksilver, and the like fluids, by the division 
of their parts in infinitum, should be subtilized, and become mediums in 
finitely fluid, nevertheless, the resistance they would make to projected 
globes would be the same. For the resistance considered in the preceding 
Propositions arises from the inactivity of the matter; and the inactivity 
of matter is essential to bodies, and always proportional to the quantity 
of matter. By the division of the parts of the fluid the resistance arising 
from the tenacity and friction of the parts may be indeed diminished : but 
the quantity of matter will not be at all diminished by this division ; and 
if the quantity of matter be the same, its force of inactivity will be the 
same ; and therefore the resistance here spoken of will be the sanue, as being 
always proportional to that force. To diminish this resistance, the quan 
tity of matter in the spaces through which the bodies move must be dimin 
ished ; and therefore the celestial spaces, through which the globes of the 
planets and comets are perpetually passing towards all parts, with the 
utmost freedom, and without the least sensible diminution of their motion, 
must be utterly void of any corporeal fluid, excepting, perhaps, some ex 
tremely rare vapours and the rays of light. 



356 THE MATHEMATICAL PRINCIPLES [BoOK 11. 

Projectiles excite a motion in fluids as they pass through them, and this 
motion arises from the excess of the pressure of the fluid at the fore parts 
of the projectile above the pressure of the same at the hinder parts : and 
cannot be less in mediums infinitely fluid than it is in air, water, and quick 
silver, in proportion to the density of matter in each. Now this excess of 
pressure does, in proportion to its quantity, not only excite a motion in the 
fluid, but also acts upon the projectile so as to retard its motion ; and there 
fore the resistance in every fluid is as the motion excited by the projectile 
in the fluid ; and cannot be less in the most subtile aether in proportion to 
the density of that aether, than it is in air, water, and Quicksilver, in pro 
portion to the densities of those fluids. 




SECTION VIII. 

Of motion propagated through fluids. 

PROPOSITION XLI. THEOREM XXXII. 

A pressure is not propagated through a fluid in rectilinear directions 
unless ichere the particles of the fluid lie in a right line. 

If the particles a, b } c, d, e, lie in a right line, the pres 
sure may be indeed directly propagated from a to e ; but 
then the particle e will urge the obliquely posited parti 
te) cles / and g obliquely, and those particles / and g will 
not sustain this pressure, unless they be supported by the 
particles h and k lying beyond them ; but the particles 
that support them are also pressed by them ; and those particles cannot 
sustain that pressure, without being supported by, and pressing upon, those 
particles that lie still farther, as / and m, and so on in itiflnitum. There 
fore the pressure, as soon as it is propagated to particles that lie out of 
right lines, begins to deflect towards one hand and the other, and will be 
propagated obliquely in infinitum ; and after it has begun to be propagat 
ed obliquely, if it reaches more distant particles lying out of the right 
line, it will deflect again on each hand and this it will do as often as it 
lights on particles that do not lie exactly in a right line. Q.E.D. 

COR. If any part of a pressure, propagated through a fluid from a given 
point, be intercepted by any obstacle, the remaining part, which is not in 
tercepted, will deflect into the spaces behind the obstacle. This may be 
demonstrated also after the following manner. Let a pressure be propagat 
ed from the point A towards any part, and, if it be possible, in rectilinear 



SEC, Vlil.l 



OF NATURAL PHILOSOPHY. 



57 




directions ; and the obstacle 
NBCK being perforated in BC, 
let all the pressure be intercepted 
but the coniform part A PQ, pass 
ing through the circular hole BC. 
Let the cone APQ, be divided 
into frustums by the transverse 
plants, de, fg, Id. Then while 
the cone ABO, propagating the 
pressure, urges the conic frustum. 
degf beyond it on the superficies 

de, and this frustum urges the next frustum fgih on the superficies/g", and 
that frustum urges a third frustum, and so in infinitum ; it is manifest 
(by the third Law) that the first frustum defg is, by the re-action of the 
second frustum fghi, as much urged and pressed on the superficies fg, as 
it urges and presses that second frustum. Therefore the frustum degf is 
compressed on both sides, that is, between the cone Ade and the frustum 
fhig; and therefore (by Case 6, Prop. XtX) cannot preserve its figure, 
unless it be compressed with the same force on all sides. Therefore with 
the same force with which it is pressed on the superficies de,fg, it will 
endeavour to break forth at the sides df, eg ; and there (being not in the 
least tenacious or hard, but perfectly fluid) it will run out, expanding it 
self,- unless there be an ambient fluid opposing that endeavour. Therefore, 
by the effort it makes to run out, it will press the ambient fluid, at its sides 

df, eg, with the same force that it does the frustum fylti ; and therefore, 
the pressure will be propagated as much from the sides df, e~, into the 
spaces NO, KL this way and that way, as it is propagated from the sr- 
ptrficies/g- towards PQ,. QJE.D. 

PROPOSITION XLII. THEOREM XXXIII. 

All motion propagated through a fluid diverges from a rectilinear pro* 

gress into ///. unmoved spaces. 
CASE 1. Let a motion be 
propagated from the point A 
through the hole BC, and, if it 
be possible, let it proceed in the 
conic space BCQP according to 
right lines diverging from the 
point A. And let us first sup 
pose this motion to be that of 
waves in the surface of standing 
water ; and let de,fg, hi, kl, &c., 
be the tops of the several waves, 
divided from each other by as 
any intermediate valleys or hollows. Then, because the water in tht 




358 THE MATHEMATICAL PRINCIPLES [BOOK I.* 

ridges of the waves is higher than in the unmoved parts of the fluid KL, 
NO, it will run down from off the tops of those ridges, e, g, i, I, &c., d y fj 
hj k, &c., this way and that way towards KL and NO ; and because the 
water is more depressed in the hollows of the waves than in the unmoved 
parts of the fluid KL, NO, it will run down into those hollows out of those 
unmoved parts. By the first deflux the ridges of the waves will dilate 
themselves this way and that way, and be propagated towards KL and NO. 
And because the motion of the waves from A towards PQ is carried on by 
a continual deflux from the ridges of the waves into the hollows next to 
them, and therefore cannot be swifter than in proportion to the celerity of 
the descent ; and the descent of the water on each side towards KL and NO 
must be performed with the same velocity ; it follows that the dilatation 
of the waves on each side towards KL and NO will be propagated with the 
same velocity ;is the waves themselves go forward with directly from A to 
PQ,. And therefore the whole space this way and that way towards KL 
and NO will be filled by the dilated waves rfgr, shis, tklt, v/nnv, &c. 
Q.E.I). That these things are so, any one may find by making the exper 
iment in still water. 

CASE 2. Let us suppose that de, fg, hi, kl, mn, represent pulses suc 
cessively propagated from the point A through an elastic medium. Con 
ceive the pulses to be propagated by successive condensations and rarefactions 
of the medium, so that the densest part of every pulse may occupy a 
spherical superficies described about the centre A, and that equal intervals 
intervene between the successive pulses. Let the lines de, fg. hi, Id, &c.. 
represent the densest parts of the pulses, propagated through the hole BC ; 
and because the medium is denser there than in the spaces on either side 
towards KL and NO. it will dilate itself as well towards those spaces KL, 
NO, on each hand, as towards the rare intervals between the pulses ; and 
thence the medium, becoming always more rare next the intervals, and 
more dense next the pulses, will partake of their motion. And because the 
progressive motion of the pulses arises from the perpetual relaxation of the 
den?er parts towards the antecedentrnre intervals; and since the pulses will 
relax themselves on each hand towards the quiescent parts of the medium 
KL, NO, with very near the same celerity ; therefore the pulses will dilate 
themselves on all sides into the unmoved parts KL, NO, with almost the 
same celerity with which they are propagated directly from the centre A; 
and therefore will fill up the whole space KLON. Q.E.D. And we find 
the same by experience also in sounds which are heard through a mountain 
interposed ; and,*if they come into a chamber through the window, dilate 
themselves into all the parts of the room, and are heard in every corner; 
and not as reflected from the opposite walls, but directly propagated from 
the window, as far as our sense can judge. 

CASE 3 Let us suppose, lastly, that a motion of any kind is propagated 



:C. VIII.j OF NATURAL PHILOSOPHY. 369 

from A through the hole BC. Then since the cause of this propagation is 
that the parts of the medium that are near the centre A disturb and agitate 
those which lie farther from it; and since the parts which are urged are 
fluid, and therefore recede every way towards those spaces where they are 
less pressed, they will by consequence recede towards all the parts of tht 
quiescent medium; as well to the parts on each hand, as KL and NO, 
as to those right before, as PQ, ; and by this means all the motion, as soon 
as it has passed through the hole BC, will begin to dilate itself, and from 
thence, as from its principle and centre, will be propagated directly every 
way. Q.E.D. 

PROPOSITION XLIII. THEOREM XXXIV. 

Every tremulous body in an elastic medium propagates the motion of 
the pulses on every side right forward ; but in a non-elastic medium 
excites a circular motion. 

CASE. 1. The parts of the tremulous body, alternately going and return 
ing, do in going urge and drive before them those parts of the medium that 
lie nearest, and by that impulse compress and condense N them ; and in re 
turning suffer those compressed parts to recede again, and expand them 
selves. Therefore the parts of the medium that lie nearest to the tremulous 
body move to and fro by turns, in like manner as the parts of the tremulous 
body itself do ; and for the same cause that the parts of this body agitate 
these parts of the medium, these parts, being agitated by like tremors, will 
in their turn agitate others next to themselves ; and these others, agitated 
in like manner, will agitate those that lie beyond them, and so on in, infin- 
itum. And in the same manner as the lirst parts of the medium were 
condensed in going, and relaxed in returning, so will the other parts be 
condensed every time they go, and expand themselves every time they re 
turn. And therefore they will not be all going and all returning at the 
same instant (for in that case they would always preserve determined dis 
tances from each other, and there could be no alternate condensation and 
rarefaction) ; but since, in the places where they are condensed, they ap 
proach to, and, in the places where they are rarefied, recede from each other, 
therefore some of them will be going while others are returning ; and so on 
in infinitum. The parts so going, and in their going condensed, are pulses, 
by reason of the progressive motion with which they strike obstacles in 
their way; and therefore the successive pulses produced by a tremulous 
body will be propagated in rectilinear directions ; and that at nearly equal 
distances from each other, because of the equal intervals of time in which 
the body, by its several tremors produces the several pulses. And though 
the parts of the tremulous body go and return .n some certain and deter 
minate direction, yet the pulses propagated from thence through the medium 
will dilate themselves towards the sides, by the foregoing Proposition : anc 7 



360 THE MATHEMATICAL PRINCIPLES [BoOK 11 

will be propagated on all sides from that tremulous body, as from a com 
mon centre, in superficies nearly spherical and concentrical. An example 
of this we have in waves excited by shaking a finger in water, which 
proceed not only forward and backward agreeably to the motion of the 
finger, but spread themselves in the manner of concentrical circles all round 
the finger, and are propagated on every side. For the gravity of the water 
supplies the place of elastic force. 

Case 2. If the medium be not elastic, then, because its parts cannot be 
condensed by the pressure arising from the vibrating parts of the tremulous 
body, the motion will be propagated in an instant towards the parts where 
the medium yields most easily, that is ; to the parts which the tremulous 
body would otherwise leave vacuous behind it. The case is the same with 
that of a body projected in any medium whatever. A medium yielding 
to projectiles does not recede in infinitum, but with a circular motion comes 
round to the spaces which the body leaves behind it. Therefore as often 
as a tremulous body tends to any part, the medium yielding to it comes 
round in a circle to the parts which the body leaves ; and as often as the 
body returns to the first place, the medium will be driven from the place it 
came round to, and return to its original place. And though the tremulous 
bod} be not firm and hard, but every way flexible, yet if it continue of a 
given magnitude, since it cannot impel the medium by its treniors any 
where without yielding to it somewhere else, the medium receding from the 
parts of the body where it is pressed will always come round in a circle to 
the parts that yield to it. Q.E.D. 

COR. It is a mistake, therefore, to think, as some have done, that the 
agitation of the parts of flame conduces to the propagation of a pressure in 
rectilinear directions through an ambient medium. A pressure of that 
kind must be derived not from the agitation only of the parts of flame, but 
from the dilatation of the whole. 

PROPOSITION XL1V. THEOREM XXXV. 

If water ascend a/id descend alternately in the erected legs KL, MN, of 
a canal or pipe ; and a pendulum be constructed whose length between 
the point of suspension and the centre of oscillation is equal to half 
the length of the ivater in the canal ; I say, that the water will ascend 
and descend in the same times in which the pendulum oscillates. 

I measure the length of the water along the axes of the canal and its legs, 
and make it equal to the sum of those axes; and take no notice of the 
resistance of the water arising from its attrition by the sides of the canal. 
Let, therefore, AB, CD, represent the mean height of the water in both 
legs ; and when the water in the leg KL ascends to the height EF, the 
water will descend in the leg MN to the height GH. Let P be a pendulou/ 



SEC. Vlll.J OF NATURAL PHILOSOPHY. VJ61 

body, VP the thread, V the point of suspension, RPQS the cycloid whicL 




ii 



L N 

the pendulum describes, P its lowest point, PQ an arc equal to the neight 
AE. The force with which the motion of the water is accelerated and re 
tarded alternately is the excess of the weight of the water in one leg above 
the weight in the other; and, therefore, when the water in the leg KL 
ascends to EF, and in the other leg descends to GH, that force is double 
the weight of the water EABF, and therefore is to the weight of the whole 
water as AE or PQ, to VP or PR. The force also with which the body P 
is accelerated or retarded in any place, as Q, of a cycloid, is (by Cor. Prop. 
LI) to its whole weight as its distance PQ, from the lowest place P to the 
length PR of the cycloid. Therefore the motive forces of the water and 
pendulum, describing the equal spaces AE, PQ, are as the weights to be 
moved ; and therefore if the water and pendulum are quiescent at first, 
those forces will move them in equal times, and will cause them to go and 
return together with a reciprocal motion. Q.E.D. 

COR. 1. Therefore the reciprocations of the water in ascending and de 
scending are all performed in equal times, whether the motion be more or 
less intense or remiss. 

COR. 2. If the length of the whole water in the canal be of 6J feet oi 
French measure, the water will descend in one second of time, and will as- 
cond in another second, and so on by turns in infinitum ; for a pendulum 
of Sy -j such feet in length will oscillate in one second of time. 

COR. 3. But if the length of the water be increased or diminished, the 
time of the reciprocation will be increased or diminished in the subdupli- 
cate ratio of the length. 

PROPOSITION XLY. THEOREM XXXVI. 

The velocity of waves is in the subduplicate ratio of the breadths. 
This follows from the construction of the following Proposition. 

PROPOSITION XLVI. PROBLEM X. 

To find the velocity of waves. 

Let a pendulum be constructed, whose length between the point of sus 
pension and the centre of oscillation is equal to the breadth of the waves 



362 THE MATHEMATICAL PRINCIPLES (BOOK 1L 

and in the time that the pendulum will perform one single oscillation the 
waves will advance forward nearly a space equal to their breadth. 

That which I call the breadth of the waves is the transverse measure 

lying between the deepest 
part of the hollows, or the 
tops of the ridges. Let 
ABCDEF represent the surface of stagnant water ascending and descend 
ing in successive waves ; and let A, C, E, &c., be the tops of the waves ; 
and let B, D, F, &c., be the intermediate hollows. Because the motion of 
the waves is carried on by the successive ascent and descent of the water, 
so that the parts thereof, as A, C, E, &c., which are highest at one time 
become lowest immediately after ; and because the motive force, by which 
the highest parts descend and the lowest ascend, is the weight of the eleva 
ted water, that alternate ascent and descent will be analogous to the recip 
rocal motion of the water in the canal, and observe the same laws as to the 
times of its ascent and descent; and therefore (by Prop. XLIV) if the 
distances between the highest places of the waves A, C, E, and the lowest 
B, D, F, be equal to twice the length of any pendulum, the highest parts 
A, C, E, will become the lowest in the time of one oscillation, and in the 
time of another oscillation will ascend again. Therefore between the pas 
sage of each wave, the time of two oscillations will intervene ; that is, the 
wave will describe its breadth in the time that pendulum will oscillate 
twice; but a pendulum of four times that length, and which therefore is 
equal to the breadth of the waves, will just oscillate once in that time. 
Q.E.L 

COR. 1. Therefore waves, whose breadth is equal to 3 7 \ French feet, 
will advance through a space equal to their breadth in one second of time; 
and therefore in one minute will go over a space of 1S3J feet ; and in an 
hour a space of 11000 feet, nearly. 

COR. 2. And the velocity of greater or less waves will be augmented or 
diminished in the subduplicatc ratio of their breadth. 

These things are true upon the supposition that the parts of water as 
cend or descend in a right line; but, in truth, that ascent and descent is 
rather performed in a circle ; and therefore I propose the time denned by 
this Proposition as only near the truth. 

PROPOSITION XLVIL THEOREM XXX VII. 

If pulses are propagated through a fluid, the .ve eral particles of the 
Jluid, goittff and returning with the shortest reciprocal motion, are al 
ways accelerated or retarded according to the law of the oscillating 
pendulum. 
Let AB, BC, CD, &c., represent equal distances of successive pulses, 

ABC the line of direction of the motion of the successive pulses propagated 



SEC. VIII.] 



OF NATURAL PHILOSOPHY. 




from A to B ; E, F, G three physical points of the quiescent medium sit 
uate in the right line AC at equal distances from each other ; Ee, F/, G^, 
equal spaces of extreme shortness, through which those 
points go and return with a reciprocal motion in each vi 
bration ; e, </>, y, any intermediate places of the same points ; 
EF, FG physical lineolae, or linear parts of the medium 
lying betAveen those points, and successively transferred into 
the places t0, 0y, and ef, fg. Let there be drawn the 
right line PS equal to the right line Ee. Bisect the same 
in O, and from the centre O, with the interval OP, describe 
the circle SIPi. Let the whole time of one vibration ; with 
its proportional parts, be expounded by the whole circum- 
lerence of this circle and its parts, in such sort, that, when 
any time PH or PHS/i is completed, if there be let fall to 
PS the perpendicular HL or hi, and there 
be taken E equal to PL or PI, the physi 
cal point E may be found in e. A point, 
as E, moving acccording to this law with 
a reciprocal motion, in its going from E 
through e to e, and returning again through 
e to E, will perform its several vibrations with the same de 
grees of acceleration and retardation with those of an oscil 
lating pendulum. We are now to prove that the several 
physical points of the medium will be agitated with such a 
kind of motion. Let us suppose, then, that a medium hath 
such a motion excited in it from any cause whatsoever, and 
consider what will follow from thence. 

In the circumference PHSA let there be taken the equal 
arcs, HI, IK, or hi, ik, having the same ratio to the whole 
circumference as the equal right lines EF, FG have to BC, 
the whole interval of the pulses. Let fall the perpendicu 
lars IM, KN, or wi, kn ; then because the points E, F, G are 
successively agitated with like motions, and perform their en tire vibrations 
composed of their going and return, while the pulse is transferred from B 
to C ; if PH or PHS/t be the time elapsed since the beginning of the mo 
tion of the point E, then will PI or PHSi be the time elapsed since the 
beginning of the motion of the point F, and PK or PHSA; the time elapsed 
since the beginning of the motion of the point G; and therefore Ee, F0, 
Gy, will be respectively equal to PL, PM, PN, while the points are going, 
and to PI, Ptn, Pn, when the points are returning. Therefore ey or EG 
4- Gy Et will, when the points are going, be equal to EG LN 




364 THE MATHEMATICAL PRINCIPLES [BOOK II. 

and in their return equal to EG + In. But ey is the breadth or ex 
pansion of the part EG of the medium in the place ey ; and therefore the 
expansion of that part in its going is to its mean expansion as EG 
LN to EG; and in its return, as EG -f In or EG + LN to EG. 
Therefore since LN is to KH as IM to the radius OP, and KH to EG 
as the circumference PHSAP to BC ; that is, if we put V for the 
radius of a circle whose circumference is equal to BC the interval of the 
pulses, as OP to V and, ex cequo, LN to EG as IM to V ; the expansion 
of the part EG, or of the physical point F in the place ey, to the mean ex 
pansion of the same part in its first place EG, will be as V IM to V 
in going, and as V -f im to V in its return. Hence the elastic force of the 
point P in the place ey to its mean elastic force in the place EG is as 
11. 11, 

v fivf * v m 1 ^ s S om o> an< ^ as v i ^ v in lts re ^ u rn. And by 

V J.1VJL V V -f Iffl V 

the same reasoning the elastic forces of the physical points E and G in going 
are as . qr and ^ ==~ to T, ; and the difference of the forces to the 



mean elastic force of the medium as T ^ 



VV-V X HL-Vx KN + HL X KN 

1 HL KN 1 

to ~ ; that is, as : to ^, or as HL KN to V ; if we suppose 

(by reason of the very short extent of the vibrations) HL and KN to be 
indefinitely less than the quantity V. Therefore since the quantity V is 
given, the difference of the forces is as HL KN ; that is (because HL 
KN is proportional to HK, and OM to OI or OP ; and because HK 
and OP are given) as OM ; that is, if F/ be bisected in ft, as ft</>. And 
for the same reason the difference of the elastic forces of the physical points 
e and y, in the return of the physical lineola ey, is as ftr/>. But that dif 
ference (that is, the excess of the elastic force of the point e above the 
elastic force of the point y) is the very force by which the intervening phy 
sical lineola ey of the medium is accelerated in going, and retarded in re 
turning ; and therefore the accelerative force of the physical lineola ey is 
as its distance from ft, the middle place of the vibration. Therefore (by 
Prop. XXXVIII, Book 1) the time is rightly expounded by the arc PI ; 
and the linear part of the medium sy is moved according to the law above- 
mentioned, that is, according to the law of a pendulum oscillating ; and 
the case is the same of all the linear parts of which the whole medium is 
compounded. Q,.E.D. 

COR. Hence it appears that the number of the pulses propagated is the 
same with the number of the vibrations of the tremulous body, and is not 
multiplied in their progress. For the physical lineola ey as soon as it 
returns to its first place is at rest ; neither will it move again, unless ii 



SEC. V11I.J OF NATURAL PHILOSOPHY. 36 

receives a new motion either from the impulse of the tremulous body, or 
of the pulses propagated from that body. As soon, therefore, as the pulses 
cease to be propagated from the tremulous body, it will return to a state 
of rest, and move no more. 

PROPOSITION XLVIII. THEOREM XXXVIII. 

The velocities of pulses propagated in an elastic fluid are in a ratin 
compounded of the subduplicate, ratio of the elastic force directly, and 
the subduplicate ratio of the density inversely ; supposing the elastic 
Jorce of the fluid to be proportional to its condensation 
CASE I. If the mediums be homogeneous, and the distances of the pulses 
in those mediums be equal amongst themselves, but the motion in one me 
dium is more intense than in the other, the contractions and dilatations of 
the correspondent parts will be as those motions ; not that this proportion 
is perfectly accurate. However, if the contractions and dilatations are not 
exceedingly intense, the error will not be sensible ; and therefore this pro 
portion may be considered as physically exact. Now the motive elastic 
forces are as the contractions and dilatations ; and the velocities generated 
in the same time in equal parts are as the forces. Therefore equal and 
corresponding parts of corresponding pulses will go and return together, 
through spaces proportional to their contractions and dilatations, with ve 
locities that are as those spaces ; and therefore the pulses, which in the 
time of one going and returning advance forward a space dq aal to their 
breadth, and are always succeeding into the places of the pulses that im 
mediately go before them, will, by reason of the equality of the distances, 
go forward in both mediums with equal velocity. 

CASE 2. If the distances of the pulses or their lengths are greater in one 
medium than in another, let us suppose that the correspondent parts de 
scribe spaces, in going and returning, each time proportional to the breadths 
of the pulses ; then will their contractions and dilatations be equal : and 
therefore if the mediums are homogeneous, the motive elastic forces, which 
agitate them with a reciprocal motion, will be equal also. Now the matter 
to be moved by these forces is as the breadth of the pulses ; and the space 
through which they move every time they go and return is in the same 
ratio. And, moreover, the time of one going and returning is in a ratic 
compounded of the subduplicate ratio of the matter, and the o-u wuupncatc 
ratio of the space ; and therefore is as the space. But the pulses advance 
a space equal to their breadths in the times of going once and returning 
once; that is, they go over spaces proportional to the times, and therefore 
are equally swift. 

CASE 3. And therefore in mediums of equal density and elastic force, 
all the pulses are equally swift. Now if the density or the elastic force of 
the medium were augmented, then, because the motive force is increased 



366 THE MATHEMATICAL PRINCIPLES [BoOK 11 

in the ratio of the elastic force, and the matter to be moved is increased in 
the ratio of the density, the time which is necessary for producing the 
same motion as before will be increased in the subduplicate ratio of the 
density, and will be diminished in the subduplicate ratio of the elastic 
force. And therefore the velocity of the pulses will be in a ratio com 
pounded of the subduplicate ratio of the density of the medium inversely, 
and the subduplicate ratio of the elastic force directly. Q,.E.D. 

This Proposition will be made more clear from the construction of the 
following Problem. 

PROPOSITION XLIX. PROBLEM XL 

The. density and elastic force of a medium being given, to find the, ve 
locity of the pulses. 

Suppose the medium to be pressed by an incumbent weight after the manner 
of our air ; and let A be the height, of a homogeneous medium, whose 
weight is equal to the incumbent weight, and whose density is the same 
with the density of the compressed medium in which the pulses are propa 
gated. Suppose a pendulum to be constructed whose length between the 
point of suspension and the centre of oscillation is A : and in the time in 
which that pendulum will perform one entire oscillation composed of 
its going and returning, the pulse will be propagated right onwards 
through a space equal to the circumference of a circle described with the 
radius A. 

For, letting those things stand which were constructed in Prop. X.LV11, 
if any physical line, as EF, describing the space PS in each vibration, be 
acted on in the extremities P and S of every going and return that it 
makes by an elastic force that is equal to its weight, it will perform its 
several vibrations in the time in which the same might oscillate in a cy 
cloid whose whole perimeter is equal to the length PS ; and that because 
equal forces will impel equal corpuscles through equal spaces in the same 
or equal times. Therefore since the times of the oscillations are in the 
subduplicate ratio of the lengths of the pendulums, and the length of the 
pendulum is equal to half the arc of the whole cycloid, the time of one vi 
bration would be to the time of the oscillation of a pendulum whose length 
is A in the subduplicate ratio of the length ^PS or PO to the length A. 
But the elastic force with which the physical lineola EG is urged, when it 
Is found in its extreme places P, S, was (in the demonstration of Prop. 
XLVII) to its whole elastic force as HL KN to V, that is (since the 
point K now falls upon P), as HK to V: and all that force, or which is 
the same thing, the incumbent weight by which the lineola EG is com 
pressed, is to the weight of the lineola as the altitude A of the incumbent 
weight to EG the length of the lineola ; and therefore, ex ctquo, the force 



SEC. VII1.I 



OF NATURAL PHILOSOPHY. 



367 



with which the lincola EG is urged in the places P and S 
is to the weight of that lineola as HK X A to V X EG ; or 
as PO X A to VV; because HK was to EG as PO to V. 
Therefore since the times in which equal bodies are impelled 
through equal spaces are reciprocally in the subduplicate 
ratio of the forces, the time of one vibration, produced by 
the action of that elastic force, will be to the time of a vi 
bration, produced by. the impulse of the weight in a subdu 
plicate ratio of VV to PO X A, and therefore to the time 
of the oscillation of a pendulum whose length is A in the 
subduplicate ratio of VV to PO X A, and the subdupli 
cate ratio of PO to A conjunctly ; that is, in the entire ra 
tio of V to A. But in the time of one 
vibration composed of the going and re 
turning of the pendulum, the pulse will 
be propagated right onward through a 
space equal to its breadth BC. There 
fore the time in which a pulse runs over 





the space BC is to the time of one oscillation composed of 
the going and returning of the pendulum as V to A, that is, 
as BC to the circumference of a circle whose radius is A. 
But the time in which the pulse will run over the space BC 
is to the time in which it will run over a length equal to 
that circumference in the same ratio; and therefore in the 
time of such an oscillation the pulse will run over a length 
equal to that circumference. G,.E.D. 

COR. 1. The velocity of the pulses is equal to that which 
heavy bodies acquire by falling with an equally accele 
rated motion, and in their fall describing half the alti 
tude A. For the pulse will, in the time of this fall, sup 
posing it to move with the velocity acquired by that fall, run over a 
space that will be equal to the whole altitude A ; and therefore in the 
time of one oscillation composed of one going and return, will go over a 
space equal to the circumference of a circle described with the radius A ; 
for the time of the fall is to the time of oscillation as the radius of a circle 
to its circumference. 

COR. 2. Therefore since that altitude A is as the elastic force of the 
fluid directly, and the density of the same inversely, the velocity of the 
pulses will be in a ratio compounded of the subduplicate ratio of the den 
sity inversely, and the subduplicate ratio of the clastic force directly. 



368 THE MATHEMATICAL PRINCIPLES |BoOK IL 



PROPOSITION L. PROBLEM XII. 

To find the distances of the pulses. 

Let the number of the vibrations of the body, by whose tremor the pulses 
are produced; be found to any given time. By that number divide the 
space which a pulse can go over in the same time, and the part found will 
be the breadth of one pulse. Q.E.I. 

SCHOLIUM. 

The last Propositions respect the motions of light and sounds ; for since 
light is propagated in right lines, it is certain that it cannot consist in ac 
tion alone (by Prop. XLI and XLIl). As to sounds, since they arise from 
tremulous bodies, they can be nothing else but pulses of the air propagated 
through it (by Prop. XLIII) ; and this is confirmed by the tremors which 
sounds, if they be loud and deep, excite in the bodies near them, as we ex 
perience in the sound of drums ; for quick and short tremors are less easily 
excited. But it is well known that any sounds, falling upon strings in 
unison with the sonorous bodies, excite tremors in those strings. This is 
also confirmed from the velocity of sounds; for since the specific gravities 
of rain-water and quicksilver are to one another as about 1 to 13f, and 
when the mercury in the barometer is at the height of 30 inches of our 
measure, the specific gravities of the air and of rain-water are to one 
another as about 1 to 870, therefore the specific gravity of air and quick 
silver are to each other as 1 to 11890. Therefore when the height of 
the quicksilver is at 30 inches, a height of uniform air, whose weight would 
be sufficient to compress our air to the density we find it to be of, must be 
equal to 356700 inches, or 29725 feet of our measure ; and this is that 
very height of the medium, which I have called A in the construction of 
the foregoing Proposition. A circle whose radius is 29725 feet is 186768 
feet in circumference. And since a pendulum 39} inches in length com 
pletes one oscillation, composed of its going and return, in two seconds of 
time, as is commonly known, it follows that a pendulum 29725 feet, or 
356700 inches in length will perform a like oscillation in 190f seconds. 
Therefore in that time a sound will go right onwards 186768 feet, and 
therefore in one second 979 feet. 

But in this computation we have made no allowance for the crassitude 
of the solid particles of the air, by which the sound is propagated instan 
taneously. Because the weight of air is to the weight of water as 1 tc 
870, and because salts are almost twice as dense as water ; if the particles 
of air are supposed to be of near the same density as those of water or salt, 
and the rarity of the air arises from the intervals of the particles ; the 
diameter of one particle of air will be to the interval between the centres 



SEC. VIIL] OF NATURAL PHILOSOPHY. 369 

of the particles as 1 to about 9 or 10, and to the interval between the par 
ticles themselves as 1 to 8 or 9. Therefore to 979 feet, which, according to 
the above calculation, a sound will advance forward in one second of time, 
\ve may add ^- 9 -, or about 109 feet, io compensate for the cra-ssitude of the 
particles of the air : and then a sound will go forward about 1088 feet in 
one second of time. 

Moreover, the vapours floating in the air being of another spring, and a 
different tone, will hardly, if at all, partake of the motion of the true air 
in which the sounds are propagated. Now if these vapours remain unmov 
ed, that motion will be propagated the swifter through the true air alone, 
and that in the subduplicate ratio of the defect of the matter. So if the 
atmosphere consist of ten parts of true air and one part of vapours, the 
motion of sounds will be swifter in the subduplicate ratio of 11 to 10, or 
very nearly in the entire ratio of 21 to 20, than if it were propagated 
through eleven parts of true air : and therefore the motion of sounds above 
discovered must be increased in that ratio. By this means the sound will 
pass through 1 142 feet in one second of time. 

These things will be found true in spring and autumn, when the air is 
rarefied by the gentle warmth of those seasons, and by that means its elas 
tic force becomes somewhat more intense. But in winter, when the air is 
condensed by the cold, and its elastic force is somewhat remitted, the mo 
tion of sounds will be slower in a subduplicate ratio of the density ; and, 
on the other hand, swifter in the summer. 

Now by experiments it actually appears that sounds do really advance 
in one second of time about 1142 feet of English measure, or 1070 feet of 
French measure. 

The velocity of sounds being known, the intervals of the pulses are known 
also. For M. Sauveur, by some experiments that he made, found that an 
open pipe about five Paris feet in length gives a sound of the same tone 
with a viol-string that vibrates a hundred times in one second. Therefore 
there are near 10J pulses in a space of 1070 Paris feet, which a sound runs 
over in asecond of time ; and therefore one pulse fills up a space of about 1 T 7 - 
Paris feet, that is, about twice the length of the pipe. From whence it is 
probable that the breadths of the pulses, in all sounds made in open pipes, 
are equal to twice the length of the pipes. 

Moreover, from the Corollary of Prop. XLVIt appears the reason why 
the sounds immediately cease with the motion of the sonorous body, and 
why they are heard no longer when we are at a great distance from the 
sonorous bodies than when we are very near them. And besides, from the 
foregoing principles, it plainly appears how it comes to pass that sounds are 
so mightily increased in speaking-trumpets ; for all reciprocal motion usea 
to be increased by the generating cause at each return. And in tubes hin 
dering the dilatation of the sounds, the motion decays more slowly, and 

24 



370 



THE MATHEMATICAL PRINCIPLES 



[BOOK II. 



recurs more forcibly ; and therefore is the more increased by the new mo 
tion impressed at each return. And these are the principal phasr. )mena oi 
sounds. 



SECTION IX. 

Of the circular motion of fluids. 

HYPOTHESIS. 

The resistance arising from the want of lubricity in the parts of a fluid, 
is, casteris paribus, proportional to the velocity with which the parts of 
the fluid are separated fro?n each other. 

PROPOSITION LI. THEOREM XXXIX. 

If a solid cylinder infinitely long, in an uniform and infinite fluid, revolve 
with an uniform motion about an axis given in position, and the fluid 
be forced round by only this impulse of the cylinder, and every part 
of the fluid persevere uniformly in its motion ; I say, that the periodic 
times of the parts of the fluid are as their distances Jrom the axis of 
the cylinder. 

Let AFL be a cylinder turning uni 
formly about the axis S, arid let the 
concentric circles BGM, CHN, DIO, 
EKP, &c., divide the fluid into innu 
merable concentric cylindric solid orbs 
of the same thickness. Then, because 
the fluid is homogeneous, the impres 
sions which the contiguous orbs make 
upon each other mutually will be (by 
the Hypothesis) as their translations 
from each, other, and as the contiguous 
superficies upon which the impressions 
are made. If the impression made upon any orb be greater or less on its 
concave than on its convex side, the stronger impression will prevail, and 
will either accelerate or retard the motion of the orb, according as it agrees 
with, or is contrary to, the motion of the same. Therefore, that every orb 
may persevere uniformly in its motion, the impressions made on both sides 
must be equal and their directions contrary. Therefore since the impres 
sions are as the contiguous superficies, and as their translations from one 
another, the translations will be inversely as the superficies, that is, inversely 
as the distances of the superficies from the axis. But the differences of 




SEC. IX] OF NATURAL PHILOSOPHY. 371 

the angular motions about the axis are as those translations applied to the 
distances, or as the translations d.rectly arid the distances inversely ; that 
is, joining these ratios together, as the squares of the distances inversely. 
Therefore if there be erected the lines A", B&, Cc, !.)</, Ee, &c., perpendic 
ular to the several parts of he infinite right line SABCDEQ,, and recip 
rocally proportional to the squares of SA, SB, SO, SO, SE, &c., and 
through the extremities of those perpendiculars there be supposed to pass 
an hyperbolic curve, the sums of the differences, that is, the whole angular 
motions, will be as the correspondent sums of the lines Ati, B6, Cc 1 , DC/, Ed?, 
that is (if to constitute a medium uniformly fluid the number of the orbs 
be increased and their breadth diminished in infinitum\ as the hyperbolic 
areas AaQ, B6Q,, CcQ,, Dc/Q,, EeQ, &c., analogous to the sums ; and the 
times, reciprocally proportional to the angular motions, will be also recip 
rocally proportional to those areas. Therefore the periodic time of any 
particle as I), is reciprocally as the area Dc/Q,, that is (as appears 
from the known methods of quadratures of curves), directly as the dis 
tance SD. Q.E.D. 

COR. 1. Hence the angular motions of the particles of the fluid are re 
ciprocally as their distances from the axis of the cylinder, and the absolute 
velocities are equal. 

COR. 2. If a fluid be contained in a cylindric vessel of an infinite length, 
and contain another cylinder within, and both the cylinders revolve about 
one common axis, and the times of their revolutions be as their semi- 
diameters, and every part of the fluid perseveres in its motion, the peri 
odic times of the several parts will be as the distances from the axis of the 
cylinders. 

COR. 3. If there be added or taken away any common quantity of angu 
lar motion from the cylinder and fluid moving in this manner; yet because 
this new motion will not alter the mutual attrition of the parts of the fluid, 
the motion of the parts among themselves will not be changed; for the 
translations of the parts from one another depend upon the attrition. 
Any part will persevere in that motion, which, by the attrition made 
on both sides with contrary directions , is no more accelerated than it is re 
tarded. 

COR. 4. Therefore if there be taken away from this whole system of the 
cylinders and the fluid all the angular motion of the outward cylinder, we 
shall have the motion of the fluid in a quiescent cylinder. 

COR. 5. Therefore if the fluid and outward cylinder are at rest, and the 
inward cylinder revolve uniformly, there will be communicated a circular 
motion to the fluid, which will be propagated by degrees through the whole 
fluid ; and will go on continually increasing, till such time as the several 
parts of the fluid acquire the motion determined in Cor. 4. 

COR. 6. And because the fluid endeavours to propagate its motion stil! 



372 



THE MATHEMATICAL PRINCIPLES 



[BOOK 11. 

farther, its impulse will carry the outmost cylinder also about with it, Tin- 
less the cylinder be violently detained; and accelerate its motion till the 
periodic times of both cylinders become equal among themselves. But if 
the outward cylinder be violently detained, it will make an effort to retard 
the motion of the fluid ; and unless the inward cylinder preserve that mo 
tion by means of some external force impressed thereon, it will make it 
3ease by degrees. 

All these things will be found true by making the experiment in deep 
standing water. 

PROPOSITION LIL THEOREM XL. 

If a solid sphere, in an uniform and infinite fluid, revolves about an axis 
given in position with an uniform motion., and thejiuid be forced round 
by only this impulse of the sphere ; and every part of the fluid perse 
veres uniformly in its motion ; I say, that the periodic times of the 
parts of the fluid are as the squares of their distances from the centre 
of the sphere. 

CASE 1. Let AFL be a sphere turn 
ing uniformly about the axis S, and let 
the concentric circles BGM, CHN, DIO, 
EKP, &c v divide the fluid into innu 
merable concentric orbs of the same 
thickness. Suppose those orbs to be 
solid ; and, because the fluid is homo 
geneous, the impressions which the con 
tiguous orbs make one upon another 
will be (by the supposition) as their 
translations from one another, and the 
contiguous superficies upon which the 
impressions are made. If the impression upon any orb be greater or less 
upon its concave than upon its convex side, the more forcible impression 
will prevail, and will either accelerate or retard the velocity of the orb, ac 
cording as it is directed with a conspiring or contrary motion to that of 
the orb. Therefore that every orb may persevere uniformly in its motion, 
it is necessary that the impressions made upon both sides of the orb should 
be equal, and have contrary directions. Therefore since the impressions 
are as the contiguous superficies, and as their translations from one another^ 
the translations will be inversely as the superficies, that is, inversely as the 
squares of the distances of the superficies from the centre. But the differ 
ences of the angular motions about the axis are as those translations applied 
to the distances, or as the translations directly and the distances inversely; 
that is, by compounding those ratios, as the cubes of the distances inversely. 
Therefore if upon the several parts of the infinite right line SABCDEQ 




SEC. IX.j OF NATURAL PHILOSOPHY. 373 

there be erected the perpendiculars Aa, B6. Cc, Dd, Ee, c. ; reciprocally 
proportional to the cubes of SA 5 SB, SO, SD, SE, etc., the sums of the 
differences, that is, the whole angular motions will be as the corresponding 
sums of the lines A#, B&, Cc, DC/, Ee, <fcc., that is (if to constitute an uni 
formly fluid medium the number of the orbs be increased and their thick 
ness diminished in infinitum), as the hyperbolic areas AaQ, B&Q,, CcQ, 
Dtf Q,, EeQ,, etc., analogous to the sums ; and the periodic times being re 
ciprocally proportional to the angular motions, will be also reciprocally 
proportional to those areas. Therefore the periodic time of any orb DIO 
is reciprocally as the area Dt/Q,, that is (by the known methods of quadra 
tures), directly as the square of the distance SD. Which was first to be 
demonstrated. 

CASE 2. From the centre of the sphere let there be drawn a great num 
ber of indefinite right lines, making given angles with the axis, exceeding 
one another by equal differences ; and, by these lines revolving about the 
axis, conceive the orbs to be cut into innumerable annuli; then will every 
annulus have four annuli contiguous to it, that is, one on its inside, one on 
its outside, and two on each hand. Now each of these annuli cannot be 
impelled equally and with contrary directions by the attrition of the inte 
rior and exterior annuli, unless the motion be communicated according to 
the law which we demonstrated in Case 1. This appears from that dem 
onstration. And therefore any series of annuli, taken in any right line 
extending itself in infinitum from the globe, will move according to the 
law of Case 1, except we should imagine it hindered by the attrition of the 
annuli on each side of it. But now in a motion, according to this law, no 
such is, and therefore cannot be, any obstacle to the motions persevering 
according to that law. If annuli at equal distances from the centre 
revolve either more swiftly or more slowly near the poles than near the 
ecliptic, they will be accelerated if slow, and retarded if swift, by their 
mutual attrition; and so the periodic times will continually approach to 
equality, according to the law of Case 1. Therefore this attrition will not 
at all hinder the motion from going on according to the law of Case 1 , and 
therefore that law will take place ; that is, the periodic times of the several 
annuli will be as the squares of their distances from the centre of the globe. 
Which was to be demonstrated in the second place. 

CASE 3. Let now every annulus be divided by transverse sections into 
innumerable particles constituting a substance absolutely and uniformly 
fluid ; and because these sections do not at all respect the law of circular 
motion, but only serve to produce a fluid substance, the law of circular mo 
tion will continue the same as before. All the very small annuli will eithei 
not at all change their asperity and force of mutual attrition upon account 
of these sections, or else they will change the same equally. Therefore the 
proportion of the causes remaining the same, the proportion of the effects 



3r4 THE MATHEMATICAL PRINCIPLES [BOOK II. 

will remain the same also ; that is, the proportion of the motions and tin 
periodic times. Q.E.D. But now as the circular motion, and the centri 
fugal force thence arising, is greater at the ecliptic than at the poles, there 
must be some cause operating to retain the several particles in their ciicles ; 
otherwise the matter that is at the ecliptic will always recede from the 
centre, and come round about to the poles by the outside of the vortex, 
and from thence return by the axis to the ecliptic with a perpetual circu 
lation. 

COR. 1. Hence the angular motions of the parts of the fluid about the 
axis of the globe are reciprocally as the squares of the distances from the 
centre of the globe, and the absolute velocities are reciprocally as the same 
squares applied to the distances from the axis. 

COR. 2. If a globe revolve with a uniform motion about an axis of a 
given position in a similar and infinite quiescent fluid with an uniform 
motion, it will communicate a whirling motion to the fluid like that of a 
vortex, and that motion will by degrees be propagated onward in infinitnm ; 
and this motion will be increased continually in every part of the fluid, till 
the periodical times of the several parts become as the squares of the dis 
tances from the centre of the globe. 

COR. 3. Because the inward parts of the vortex are by reason of their 
greater velocity continually pressing upon and driving forward the external 
parts, and by that action are perpetually communicating motion to them, 
and at the same time tho se exterior parts communicate the same quantity 
of motion to those that lie still beyond them, and by this action preserve 
the quantity of their motion continually unchanged, it is plain that the 
motion is perpetually transferred from the centre to the circumference of 
the vortex, till it is quite swallowed up and lost in the boundless extent of 
that circumference. The matter between any two spherical superficies 
concentrical to the vortex will never be accelerated ; because that matter 
will be always transferring the motion it receives from the matter nearer 
the centre to that matter which lies nearer the circumference. 

COR. 4. Therefore, in order to continue a vortex in the same state of 
motion, some active principle is required from which the globe may receive 
continually the same quantity of motion which it is always communicating 
to the matter of the vortex. Without such a principle it will undoubtedly 
come to pass that the globe and the inward parts of the vortex, being al 
ways propagating their motion to the outward parts, and not receiving any 
new motion, will gradually move slower and slower, and at last be carried 
round no longer. 

COR. 5. If another globe should be swimming in the same vortex at a 
certain distance from its centre, and in the mean time by some force revolve 
constantly about an axis of a given inclination, the motion of Jiis globe 
will drive the fluid round after the manner of a vortex and at first this 



SEC. IX.] OF NATURAL PHILOSOPHY. 375 

new and small vortex will revolve with its globe about the centre of the 
other; and in the mean time its motion will creep on farther and farther, 
and by degrees be propagated in iiifinitum, after the manner of the first 
vortex. And for the same reason that the globe of the new vortex wat 
carried about before by the motion of the other vortex, the globe of this 
other will be carried about by the motion of this new vortex, sc that the 
two globes will revolve about some intermediate point, and by reason of 
that circular motion mutually fly from each other, unless some force re 
strains them. Afterward, if the constantly impressed forces, by which the 
globes persevere in their motions, should cease, and every thing be left to 
act according to the laws of mechanics, the motion of the globes will lan 
guish by degrees (for the reason assigned in Cor. 3 arid 4), and the vortices 
at last will quite stand still. 

COR. 6. If several globes in given places should constantly revolve with 
determined velocities about axes given in position, there would arise from 
them as many vortices going on in infinitum. For upon the same account 
that any one globe propagates its motion in itifinitum, each globe apart 
will propagate its own motion in infiidtwtn also ; so that every part of the 
infinite fluid will be agitated with a motion resulting from the actions of 
all the globes. Therefore the vortices will not be confined by any certain 
limits, but by degrees run mutually into each other ; and by the mutual 
actions of the vortices on each other, the globes will be perpetually moved 
from their places, as was shewn in the last Corollary ; neither can they 
possibly keep any certain position among themselves, unless some force re 
strains them. But if those forces, which are constantly impressed upon 
the globes to continue these motions, should cease, the matter (for the rea 
son assigned in Cor. 3 and 4) will gradually stop, and cease to move in 
vortices. 

COR. 7. If a similar fluid be inclosed in a spherical vessel, and, by the 
uniform rotation of a globe in its centre, is driven round in a vortex ; and 
the globe and vessel revolve the same way about the same axis, and their 
periodical times be as the squares of the semi-diameters ; the parts of the 
fluid will not go on in their motions without acceleration or retardation, 
till their periodical times are as the squares of their distances from 
the centre of the vortex. No constitution of a vortex can be permanent 
but this. 

COR. 8. If the vessel, the inclosed fluid, and the globe, retain this mo 
tion, and revolve besides with a common angular motion about any given 
axis, because the mutual attrition of the parts of the fluid is not changed 
by this motion, the motions of the parts among each other will not be 
changed ; for the translations of the parts among themselves depend upon 
this attrition. Any part will persevere in that motion in which its attri- 



376 THE MATHEMATICAL PRINCIPLES [BOOK II. 

tion on one side retards it just as much as its attrition on the other side 
accelerates it. 

COR. 9. Therefore if the vessel be quiescent, and the motion of the 
globe be given, the motion of the fluid will be given. For conceive a plane 
to pass through the axis of the globe, and to revolve with a contrary mo 
tion ; and suppose the sum of the time of this revolution and of the revolu 
tion of the globe to be to the time of the revolution of the globe as the 
square of the semi-diameter of the vessel.to the square of the semi-diameter 
of the globe ; and the periodic times of the parts of the fluid in respect of 
this plane will be as the squares of their distances from the centre of the 
globe. 

COR. 10. Therefore if the vessel move about the same axis with the globe, 
or with a given velocity about a different one, the motion of the fluid will 
be given. For if from the whole system we take away the angular motion 
of the vessel, all the motions will remain the same among themselves as 
before, by Cor. 8, and those motions will be given by Cor. 9. 

COR. 11. If the vessel and the fluid are quiescent, and the globe revolves 
with an uniform motion, that motion will be propagated by degrees through 
the whole fluid to the vessel, and the vessel will be carried round by it, 
unless violently detained ; and the fluid and the vessel will be continually 
accelerated till their periodic times become equal to the periodic times of 
the globe. If the vessel be either withheld by some force, or revolve with 
any constant and uniform motion, the medium will come by little and 
little to the state of motion defined in Cor. 8, 9, 10, nor will it ever perse 
vere in any other state. But if then the forces, by which the globe and 
vessel revolve with certain motions, should cease, and the whole system be 
left to act according to the mechanical laws, the vessel and globe, by means 
of the intervening fluid, will act upon each other, and will continue to 
propagate their motions through the fluid to each other, till their periodic 
times become equal among themselves, and the whole system revolves to 
gether like one solid body. 

SCHOLIUM. 

In all these reasonings I suppose the fluid to consist of matter of uniform 
density and fluidity ; I mean, that the fluid is such, that a globe placed 
any where therein may propagate with the same motion of its own, at dis 
tances from itself continually equal, similar and equal motions in the fluid 
in the same interval of time. The matter by its circular motion endeavours 
to recede from the axis of the vortex, and therefore presses all the matter 
that lies beyond. This pressure makes the attrition greater, and the 
Separation of the parts more difficult ; and by consequence diminishes 
the fluidity of the matter. Again ; if the parts of the fluid are in any one 
place denser or larger than in the others, the fluidity will be less in that 
[lace, because there are fewer superficies where the parts can be separated 



fclC IX.] Or NATURAL PHILOSOPHY. 3?< 

from each other. In these cases I suppose the defect of the fluidity to be 
supplied by the smoothness or softness of the parts, or some other condi 
tion ; otherwise the matter where it is less fluid will cohere more, and be 
more sluggish, and therefore will receive the motion more slowly, and pro 
pagate it farther than agrees with the ratio above assigned. If the vessel 
be riot spherical, the particles will move in lines not circular, but answer 
ing to the figure of the vessel ; and the periodic times will be nearly as the 
squares of the mean distances from the centre. In the parts between the 
centre and the circumference the motions will be slower where the spaces 
are wide, and swifter where narrow ; but yet the particles will not tend to the 
circumference at all the more for their greater swiftness ; for they then 
describe arcs of less curvity, and the conatus of receding from the centre is 
as much diminished by the diminution of this curvature as it is augment 
ed by the increase of the velocity. As they go out of narrow into wide 
spaces, they recede a little farther from the centre, but in doing so are re 
tarded ; and when they come out of wide into narrow spaces, they are again 
accelerated ; and so each particle is retarded and accelerated by turns for 
ever. These things will come to pass in a rigid vessel ; for the state of 
vortices in an infinite fluid is known by Cor. 6 of this Proposition. 

I have endeavoured in this Proposition to investigate the properties of 
vortices, that I might find whether the celestial phenomena can be explain 
ed by them; for the phenomenon is this, that the periodic times of the 
planets revolving about Jupiter are in the sesquiplicate ratio of their dis 
tances from Jupiter s centre ; and the same rule obtains also among the 
planets that revolve about the sun. And these rules obtain also with the 
greatest accuracy, as far as has been yet discovered by astronomical obser- 
tion. Therefore if those planets are carried round in vortices revolving 
about Jupiter and the sun, the vortices must revolve according to that 
law. But here we found the periodic times of the parts of the vortex to 
be in the duplicate ratio of the distances from the centre of motion ; and 
this ratio cannot be diminished and reduced to the sesquiplicate, unless 
either the matter of the vortex be more fluid the farther it is from the cen 
tre, or the resistance arising from the want of lubricity in the parts of the 
fluid should, as the velocity with which the parts of the fluid are separated 
goes on increasing, be augmented with it in a greater ratio than that in 
which the velocity increases. But neither of these suppositions seem rea 
sonable. The more gross and less fluid parts will tend to the circumfer 
ence, unless they are heavy towards the centre. And though, for the sake 
of demonstration, I proposed, at the beginning of this Section, an Hypoth 
esis that the resistance is proportional to the velocity, nevertheless, it is in 
truth probable that the resistance is in a less ratio than that of the velo 
city ; which granted, the periodic times of the parts of the vortex will be 
in a greater than the duplicate ratio of the distances from its centre. If, 
as some think, the vortices move more swiftly near the centre, then slower 



378 THE MATHEMATICAL PRINCIPLES [BOOK IT 

to a certain limit, then again swifter near the circumference, certainly 
neither the sesquiplicate, nor any other certain and determinate ratio, can 
obtain in them. Let philosophers then see how that phenomenon of the 
sesquiplicate ratio can be accounted for by vortices. 

PROPOSITION LIII. THEOREM XLI. 

Bodies carried about in a vortex, and returning- in the same orb, are of 
the same density with the vortex, and are moved according to the 
same law with the parts of the vortex, as to velocity and direction oj 
motion. 

For if any small part of the vortex, whose particles or physical points 
preserve a given situation among each other, be supposed to be congealed, 
this particle will move according to the same law as before, since no change 
is made either in its density, vis insita, or figure. And again ; if a congealed 
or solid part of the vortex be of the same density with the rest of the vortex, 
and be resolved into a fluid, this will move according to the same law as 
before, except in so far as its particles, now become fluid, may be moved 
among themselves. Neglect, therefore, the motion of the particles among 
themselves as not at all concerning the progressive motion of the whole, and 
the motion of the whole will be the same as before. But this motion will be 
the same with the motion of other parts of the vortex at equal distances 
from the centre; because the solid, now resolved into a fluid, is become 
perfectly like to the other parts of the vortex. Therefore a solid, if it be 
of the same density with the matter of the vortex, will move with the same 
motion as the parts thereof, being relatively at rest in the matter that sur 
rounds it. If it be more dense, it will endeavour more than before to re 
cede from the centre ; and therefore overcoming that force of the vortex, 
by which, being, as it were, kept, in equilibrio, it was retained in its orbit, 
it will recede from the centre, and in its revolution describe a spiral, re 
turning no longer into the same orbit. And, by the same argument, if it 
be more rare, it will approach to the centre. Therefore it can never con 
tinually go round in the same orbit, unless it be of the same density with 
the fluid. But we have shewn in that case that it would revolve accord 
ing to the same law with those parts of the fluid that are at the same or 
equal distances from the centre of the vortex. 

COR. 1. Therefore a solid revolving in a vortex, and continually going 
round in the same orbit, is relatively quiescent in the fluid that carries it. 
COR. 2. And if the vortex be of an uniform density, the same body may 
revolve at any distance from the centre of the vortex. 

SCHOLIUM. 

Hence it is manifest that the planets are not carried round in corporeal 
vortices ; for, according to the Copernican hypothesis, the planets going 




SEC. IX.] OF NATURAL PHILOSOPHY. 379 

round the sun revolve in ellipses, having the sun in their common focus ; 
and by radii drawn to the sun describe 
areas proportional to the times. But 
now the parts of a vortex can never re 
volve with such a motion. Let AD, 
BE, CF, represent three orbits describ 
ed about the sun S, of which let the 
utmost circle CF be concentric to the 
sun ; and let the aphelia of the two in 
nermost be A, B j and their perihelia 
D, E. Therefore a body revolving in 
the orb CF, describing, by a radius 
drawn to the sun, areas proportional to 
the times, will move with an uniform motion. And, according to the laws 
of astronomy, the body revolving in the orb BE will move slower in its 
aphelion B, and swifter in its perihelion E ; whereas, according to the 
laws of mechanics, the matter of the vortex ought to move more swiftly in 
the narrow space between A and C than in the wide space between D and 
F ; that is, more swiftly in the aphelion than in the perihelion. Now these 
two conclusions contradict each other. So at the beginning of the sign of 
Virgo, where the aphelion of Mars is at present, the distance between the* 
orbits of Mars and Venus is to the distance between the same orbits, at the 
beginning of the sign of Pisces, as about 3 to 2 ; and therefore the matter 
of the vortex between those orbits ought to be swifter at the beginning of 
Pisces than at the beginning of Virgo in the ratio of 3 to 2 ; for the nar 
rower the space is through which the same quantity of matter passes in the 
same time of one revolution, the greater will be the velocity with which it 
passes through it. Therefore if the earth being relatively at rest in this 
celestial matter should be carried round by it, and revolve together with it 
about the sun, the velocity of the earth at the beginning of Pisces 
would be to its velocity at the beginning of Virgo in a sesquialteral ratio. 
Therefore the sun s apparent diurnal motion at the beginning of Virgo 
ought to be above 70 minutes, and at the beginning of Pisces less than 48 
minutes; whereas, on the contrary, that apparent motion of the sun is 
really greater at the beginning of Pisces than at the beginning of Virgo; 
as experience testifies ; and therefore the earth is swifter at the beginning 
of Virgo than at the beginning of Pisces ; so that the hypothesis of vor 
tices is utterly irreconcileable with astronomical phenomena, and rather 
serves to perplex than explain the heavenly motions. How these mo 
tions are performed in free spaces without vortices, may be understood 
by the first Book j and I shall now more fully treat of it in the following 
Book. 



BOOK III 



BOOK III. 



IN the preceding Books I have laid down the principles of philosophy , 
principles not philosophical, but mathematical : such, to wit, as we may 
build our reasonings upon in philosophical inquiries. These principles are 
the laws and conditions of certain motions, and powers or forces, which 
chiefly have respect to philosophy : but, lest they should have appeared of 
themselves dry and barren, I have illustrated them here and there with 
some philosophical scholiums, giving an account of such things as are of 
more general nature, and which philosophy seems chiefly to be founded on ; 
such as the density and the resistance of bodies, spaces void of all bodies, 
and the motion of light and sounds. It remains that, from the same prin 
ciples, I now demonstrate the frame of the System of the World. Upon 
this subject I had, indeed, composed the third Book in a popular method, 
that it might be read by many ; but afterward, considering that such as 
had not sufficiently entered into the principles could not easily discern the 
strength of the consequences, nor lay aside the prejudices to which they had 
been many years accustomed, therefore, to prevent the disputes which might 
be raised upon such accounts, I chose to reduce the substance of this Book 
into the form of Propositions (in the mathematical way), which should be 
read by those only who had first made themselves masters of the principles 
established in the preceding Books : not that I would advise any one to the 
previous study of every Proposition of those Books ; for they abound with 
such as might cost too much time, even to readers of good mathematical 
learning. It is enough if one carefully reads the Definitions, the Laws of 
Motion, and the first three Sections of the first Book. He may then pass 
on to this Book, and consult such of the remaining Propositions of the 
first two Books, as the references in this, and his occasions, shall require. 



384 THE MATHEMATICAL PRINCIPLES [BOOK III. 

RULES OF REASONING IN PHILOSOPHY, 



RULE I. 

We are I o admit no more causes of natural things than such as are both 

true and sufficient to explain their appearances. 

To this purpose the philosophers say that Nature does nothing in vain, 
and more is in vain when less will serve ; for Nature is pleased with sim 
plicity, and affects not the pomp of superfluous causes. 

RULE II. 

Therefore to the same natural effects we must, as far as possible, assign 

the same causes. 

As to respiration in a man and in a beast; the descent of stones in Europe 
and in America ; the light of our culinary fire and of the sun ; the reflec 
tion of light in the earth, and in the planets. 

RULE III. 

The qualities of bodies, which admit neither intension nor remission oj 
degrees, and which are found to belong to all bodies within the reach 
of our experiments, are to be esteemed the universal qualities of all 
bodies whatsoever. 

For since the qualities of bodies are only known to us by experiments, we 
are to hold for universal all such as universally agree with experiments ; 
nnd such as are not liable to diminution can never be quite taken away. 
We are certainly not to relinquish the evidence of experiments for the sake 
of dreams and vain fictions of our own devising ; nor are we to recede from 
the analogy of Nature, which uses to be simple, and always consonant to 
itself. We no other way know the extension of bodies than by our senses, 
nor do these reach it in all bodies; but because we perceive extension in 
all that are sensible, therefore we ascribe it universally to all others also. 
That abundance of bodies are hard, we learn by experience ; and because 
the hardness of the whole arises from the hardness of the parts, we therefore 
justly infer the hardness of the undivided particles not only of the bodies 
we feel but of all others. That all bodies are impenetrable, we gather not 
from reason, but from sensation. The bodies which we handle we find im 
penetrable, and thence conclude impenetrability to be an universal property 
of all bodies whatsoever. That all bodies are rnoveable, and endowed with 
certain powers (which we call the vires inertias] of persevering in their mo 
tion, or in their rest, we only infer from the like properties observed in the 



BOOK 1II.J OF NATURAL PHILOSOPHY. 385 

bodies which we have seen. The extension, hardness, impenetrability, mo 
bility, and vis inertia of the whole, result from the extension, hardness, 
impenetrability, mobility, and vires inertia of the parts; and thence we 
conclude the least particles of all bodies to be also all extended, and hard 
and impenetrable, and moveable, and endowed with their proper vires inertia. 
And this is the foundation of all philosophy. Moreover, that the divided 
but contiguous particles of bodies may be separated from one another, is 
matter of observation ; and, in the particles that remain undivided, our 
minds are able to distinguish yet lesser parts, as is mathematically demon 
strated. But whether the parts so distinguished, and not yet divided, may, 
by the powers of Nature, be actually divided and separated from one an 
other, we cannot certainly determine. Yet, had we the proof of but one 
experiment that any undivided particle, in breaking a hard and solid body, 
suffered a division, we might by virtue of this rule conclude that the un 
divided as well as the divided particles may be divided and actually sep 
arated to infinity. 

Lastly, if it universally appears, by experiments and astronomical obser 
vations, that all bodies about the earth gravitate towards the earth, and 
that in proportion to the quantity of matter which they severally contain ; 
that the moon likewise, according to the quantity of its matter, gravitates 
towards the earth ; that, on the other hand, our sea gravitates towards the 
moon ; and all the planets mutually one towards another ; and the comets 
in like manner towards the sun ; we must, in consequence of this rule, uni 
versally allow that all bodies whatsoever are endowed with a principle ot 
mutual gravitation. For the argument from the appearances concludes with 
more force for the universal gravitation of all bodies than for their impen 
etrability ; of which, among those in the celestial regions, we have no ex 
periments, nor any manner of observation. Not that I affirm gravity to be 
essential to bodies : by their vis insita I mean nothing but their vis iiicrticz. 
This is immutable. Their gravity is diminished as they recede from the 
earth. 

RULE IV. 

In experimental philosophy we are to look upon propositions collected by 
general induction from, phenomena as accurately or very nearly true, 
notwithstanding any contrary hypotheses that may be imagined, till 
such time as other phenomena occur, by which they may either be made 
more accurate, or liable to exceptions. 
This rule we must follow, that the argument of induction may not bf 

evaded by hypotheses. 

25 



386 



THE MATHEMATICAL PRINCIPLES 



[BooK III. 



PHENOMENA, OR APPEARANCES, 



PHENOMENON I. 

That the circumjovial planets, by radii drawn to Jupiter s centre, de 
scribe areas proportional to the times of description ; and that their 
periodic times, the fixed stars being at rest, are in the sesquiplicate 
proportion of their distances from, its centre. 

This we know from astronomical observations. For the orbits of these 
planets differ but insensibly from circles concentric to Jupiter ; and their 
motions in those circles are found to be uniform. And all astronomers 
agree that their periodic times are in the sesquiplicate proportion of the 
semi-diameters of their orbits; and so it manifestly appears from the fol- 
1 owing table. 

The periodic times of the satellites of Jupiter. 

H 18 h . 27 . 34". 3 d . 13 h . 13 42". 7 d . 3 1 . 42 36". 16 d . 16 h . 32 9". 
The distances of the satellites from Jupiter s centre. 



From the observations of 


1 


2 


3 


4 


Borelli 


5$ 


8| 


14 


24? 1 . 


Townly by the Microm. . . . 
Cassini by the Telescope . . . 
Cassini by the cclip. of the satel. . 


5,52 
5 
5| 


8,78 
8 
9 


13,47 
13 
14ff 


*! 1 

24,72 ] semi-diameter of 
23 Jupiter. 
25A J 


From the periodic times 


5,667 


9,017 


14,384 


25,299 



Mr. Pound has determined, by the help of excellent micrometers, the 
diameters of Jupiter and the elongation of its satellites after the following 
manner. The greatest heliocentric elongation of the fourth satellite from 
Tupiter s centre was taken with a micrometer in a 15 feet telescope, and at 
the mean distance of Jupiter from the earth was found about 8 16". The 
elongation of the third satellite was taken with a micrometer in a telescope 
of 123 feet, and at the same distance of Jupiter from the earth was found 
4 42". The greatest elongations of the other satellites, at the same dis 
tance of Jupiter from the earth, are found from the periodic times to be 2 
56" 47 ", and 1 51" 6 ". 

The diameter of Jupiter taken with the micrometer in a 123 feet tele 
scope several times, and reduced to Jupiter s mean distance from the earth, 
proved always less than 40", never less than 38", generally 39". This di 
ameter in shorter telescopes is 40", or 41"; for Jupiter s light is a little 
dilated by the unequal refrangibility of the rays, and this dilatation bears 
3 less ratio to the diameter of Jupiter in the longer and more perfect tele- 
escopes than in those which are shorter and less perfect. The times :i 



HOOK. III.] OF NATURAL PHILOSOPHY 387 

which two satellites, the first and the third, passed over Jupiter s body, were 
observed, from the beginning of the ingress to the beginning of the egress, 
and from the complete ingress to the complete egress, with the long tele 
scope. And from the transit of the first satellite, the diameter of Jupiter 
at its mean distance from the earth came forth 37 J-". and from the transit 
of the third 371". There was observed also the time in which the shadow 
of the first satellite passed over Jupiter s body, and thence the diameter of 
Jupiter at its mean distance from the earth came out about 37". Let us 
suppose its diameter to be 37}" very nearly, and then the greatest elonga 
tions of the first, second, third, and fourth satellite will be respectively 
equal to 5,965, 9,494, 15,141, and 26,63 semi-diameters of Jupiter. 

PHENOMENON II. 

Tkat the. circumsalurnal planets, by radii drawn, to Saturtfs centre, de 
scribe areas proportional to the times of description ; and that their 
periodic times, the fixed stars being at rest, are in the sesqniplicata 
proportion uf their distances from its centre. 

For, as Cassiui from his own observations has determined, theii distan 
ces from Saturn s centre and their periodic times are as follow. 

The periodic times of the satellites of Saturn. 

l d . 2l h . IS 27". 2 d . 17 h . 41 22". 4 d . 12". 25 12". 15 d . 22^. 41 14", 

79 1 . 7 1 . 48 00". 

The distances of the satellites from Saturn s centre, in semi-diameters oj 

itv ring . 

From observations li-. 2f 3|. 8. 24 

From the periodic times . . . 1,93. 2,47. 3,45. 8. 23.35. 

The greatest elongation of the fourth satellite from Saturn s centre is 
commonly determined from the observations to be eight of th-se semi- 
diameters very nearly. But the greatest elongation of this satellite from 
Saturn s centre, when taken with an excellent micrometer iuMr../fuyg en8 > 
telescope of 123 feet, appeared to be eight semi-diameters and T 7 - of a semi- 
diameter. And from this observation arid the periodic times the distances 
of the satellites from Saturn s centre in serni-diameters of the ring are 2.1. 
2,69. 3,75. 8,7. and 25,35. The diameter of Saturn observed in the same 
telescope was found to be to the diameter of the ring as 3 to 7 ; and the 
diameter of the ring, May 28-29, 1719, was found to be 43" ; and th:*nce 
the diameter of the ring when Saturn is at its mean distance from the 
earth is 42", and the diameter of Saturn 18". These things appear so in 
very long and excellent telescopes, because in such telescopes the apparent 
magnitudes of the heavenly bodies bear a greater proportion to the dilata 
tion of light in the extremities of those bodies than in shorter telescopes. 



3S8 THE MATHEMATICAL PRINCIPLES [HoOK III 

If we, then, reject all the spurious light, the diameter of Saturn will not 
amount to more than 16". 

PHENOMENON III. 



That the five primary planets, Mercury, Venus, Mars, Jupiter, and Sat 
urn, with their several orbits, encompass the sun. 
That Mercury and Venus revolve about the sun, is evident from their 
moon-like appearances. When they shine out with a full face, they are, in 
respect of us, beyond or above the sun ; when they appear half full, they 
are about the same height on one side or other of the sun ; when horned, 
they are below or between us and the sun ; and they are sometimes, when 
directly under, seen like spots traversing the sun s disk. That Mars sur 
rounds the sun, is as plain from its full face when near its conjunction with 
the sun. and from the gibbous figure which it shews in its quadratures. 
And the same thing is demonstrable of Jupiter and Saturn, from their ap 
pearing full in all situations ; for the shadows of their satellites that appear 
sometimes upon their disks make it plain that the light they shine with is 
not their own, but borrowed from the sun. 

PHENOMENON IV. 

That the fixed stars being at rest, the periodic times of the five primary 
planets, and (whether of the suit about the earth, or) of the earth about 
the sun, are in the sesquiplicate proportion of their mean distances 
from the sun. 

This proportion, first observed by Kepler, is now received by all astron 
omers ; for the periodic times are the same, and the dimensions of the orbits 
are the same, whether the sun revolves about the earth, or the earth about 
the sun. And as to the measures of the periodic times, all astronomers are 
agreed about them. But for the dimensions of the orbits, Kepler and Bul- 
lialdns, above all others, have determined them from observations with the 
greatest accuracy ; and the mean distances corresponding to the periodic 
times differ but insensibly from those which they have assigned, and for 
the most part fall in between them ; as we may see from the following table. 

The periodic times with respect to the fixed stars, of the planets and earth 
revolving about the sun. in days and decimal parts of a day. 

* ^ * $ ? * 

10759,275. 4332,514. 686,9785. 365,2565. 224,6176. 87,9692. 

The mean distances of the planets and of the earth from the sun. 

* V I 

According to Kepler 951000. 519650. 152350. 

to Bullialdus 954198. 522520. 152350. 

to the periodic times .... 954006. 520096. 152369 



BOOK III.] OF NATURAL PHILOSOPHY. 389 

J ? * 

According to Kepler 100000. 72400. 38806 

" to Bnllialdus ... . . . 100000. 72398. 38585 

" to the periodic times 100000. 72333. 38710. 

As to Mercury and Venus, there can be no doubt about their distances 
from the sun ; for they are determined by the elongations of those planets 
from the sun ; and for the distances of the superior planets, all dispute is 
cut off by the eclipses of the satellites of Jupiter. For by those eclipses 
the position of the shadow which Jupiter projects is determined ; whence 
we have the heliocentric longitude of Jupiter. And from its helio 
centric and geocentric longitudes compared together, we determine its 
distance. 

PHENOMENON V. 

Then the primary planets, by radii drawn to the earth, describe areas no 
wise proportional to the times ; but that the areas which they describe 
by radii drawn to the snn are proportional to the times of descrip 
tion. 

For to the earth they appear sometimes direct, sometimes stationary, 
nay, and sometimes retrograde. But from the sun they are always seen 
direct, and to proceed with a motion nearly uniform, that is to say, a little 
swifter in the perihelion and a little slower in the aphelion distances, so as 
to maintain an equality in the description of the areas. This a noted 
proposition among astronomers, and particularly demonstrable in Jupiter, 
from the eclipses of his satellites; by the help of which eclipses, as we have 
said, the heliocentric longitudes of that planet, and its distances from the 
sun, are determined. 

PHENOMENON VI. 

That the moon, by a radius drawn to the earths centre, describes an area 

proportional to the time of description. 

This we gather from the apparent motion of the moon, compared with 
its apparent diameter. It is true that the motion of the moon is a little 
disturbed by the action of the sun : but in laying down these Phenomena 
I neglect those imall and inconsiderable errors. 



390 THE MATHEMATICAL PRINCIPLES [BOOK III 

PROPOSITIONS- 

PROPOSITION I. THEOREM I. 

That the forces by which the circumjovial planets are continually drawn 
off from rectilinear motions, and retained in their proper orbits, tend 
to Jupiter s centre ; and are reciprocally as the squares of the distances 
of the places of those planets/ro?/i that centre. 
The former part of this Proposition appears from Pham. I, and Prop. 

II or III, Book I : the latter from Phaen. I, and Cor. 6, Prop. IV, of the same 

Book. 

The same thing we are to understand of the planets which encompass 

Saturn, by Phaon. II. 

PROPOSITION II. THEOREM II. 

That the forces by which the primary planets are continually drawn off 
from rectilinear motions, and retained in their proper orbits, tend to 

the sun. ; and are reciprocally as the squares of the distances of the 

places of those planets from the sun s centre. 

The former part of the Proposition is manifest from Phasn. V, and 
Prop. II, Book I ; the latter from Phaen. IV, and Cor. 6, Prop. IV, of the 
same Book. But this part of the Proposition is, with great accuracy, de 
monstrable from the quiescence of the aphelion points ; for a very small 
aberration from the reciprocal duplicate proportion would (by Cor. 1, Prop. 
XLV, Book I) produce a motion of the apsides sensible enough in every 
single revolution, and in many of them enormously great. 

PROPOSITION III. THEOREM III. 

That the force by which the moon is retained in its orbit tends to the 
earth ; and is reciprocally as the square of the distance of its plac>>, 
from the earths centre. 

The former part of the Proposition is evident from Pha3n. VI, and Prop. 
II or III, Book I ; the latter from the very slow motion of the moon s apo 
gee; which in every single revolution amounting but to 3 3 in conse- 
quentia, may be neglected. For (by Cor. 1. Prop. XLV, Book I) it ap 
pears, that, if the distance of the moon from the earth s centre is to the 
semi-diameter of the earth as D to 1, the force, from which such a motion 
will result, is reciprocally as D 2 ^f 3, i. e., reciprocally as the power of D, 
whose exponent is 2^^ ; that is to say, in the proportion of the distance 
something greater than reciprocally duplicate, but which comes 59f time? 
nearer to the duplicate than to the triplicate proportion. But in regard 
that this motion is owinsr to the action of the sun (as we shall afterwards 



BOOK III.] OF NATURAL PHILOSOPHY. 391 

shew), it is here to be neglected. The action of the sun, attracting the 
moon from the earth, is nearly as the moon s distance from the earth ; and 
therefore (by what we have shewed in Cor. 2, Prop. XLV. Book I) is to the 
centripetal force of the moon as 2 to 357,45, or nearly so ; that is, as 1 to 
178 f- . And if we neglect so inconsiderable a force of the sun, the re 
maining force, by which the moon is retained in its orb, will be recipro 
cally as D 2 . This will yet more fully appear from comparing this force 
with the force of gravity, as is done in the next Proposition. 

COR. If we augment the mean centripetal force by which the moon is 
retained in its orb, first in the proportion of 177%$ to 178ff, and then in 
the duplicate proportion of the semi-diameter of the earth to the mean dis 
tance of the centres of the moon and earth, we shall have the centripetal 
force of the moon at the surface of the earth ; supposing this force, in de 
scending to the earth s surface, continually to increase in the reciprocal 
duplicate proportion of the height. 

PROPOSITION IV. THEOREM IV. 

That the moon gravitates towards the earth, and by thejorce oj gravity 
is continually drawn off from a rectilinear motion, and retained in 
its orbit. 

The mean distance of the moon from the earth in the syzygies in semi- 
diameters of the earth, is, according to Ptolemy and most astronomers, 
59 : according to Vendelin and Huygens, 60 ; to Copernicus, 60 1 ; to 
Street, 60| ; and to Tycho, 56|. But Tycho, and all that follow his ta 
bles of refraction, making the refractions of the sun and moon (altogether 
against the nature of light) to exceed the refractions of the fixed stars, and 
that by four or five minutes near the horizon, did thereby increase the 
moon s horizontal parallax by a like number of minutes, that is, by a 
twelfth or fifteenth part of the whole parallax. Correct this error, and 
the distance will become about 60^ semi-diameters of the earth, near to 
what others have assigned. Let us assume the mean distance of 60 diam 
eters in the syzygies ; and suppose one revolution of the moon, in respect 
of the fixed stars, to be completed in 27 d . 7 h . 43 , as astronomers have de 
termined ; and the circumference of the earth to amount to 123249600 
Paris feet, as the French have found by mensuration. And now if we 
imagine the moon, deprived of all motion, to be let go, so as to descend 
towards the earth with the impulse of all that force by which (by Cor. 
Prop. Ill) it is retained in its orb, it will in the space of one minute of time, 
describe in its fall 15 T ^ Paris feet. This we gather by a calculus, founded 
either upon Prop. XXXVI, Book [, or (which comes to the same thing; 
upon Cor. 9, Prop. IV, of the same Book. For the versed sine of that arc, 
which the moon, in the space of one minute of time, would by its mean 



392 THE MATHEMATICAL PRINCIPLES [BOOK III 

motion describe at the distance of 60 seini-diameters of the earth, is nearly 
15^ Paris feet, or more accurately 15 feet, 1 inch, and 1 line . Where 
fore, since that force, in approaching to the earth, increases in the recipro 
cal duplicate proportion of the distance, and, upon that account, at the 
surface of the earth, is 60 X 60 times greater than at the moon, a body 
in our regions, falling with that force, ought in the space of one minute of 
time, to describe 60 X 60 X 15 T ] Paris feet; and, in the space of one sec 
ond of time, to describe 15 ,\ of those feet; or more accurately 15 feet, 1 
inch, and 1 line f . And with this very force we actually find that bodies 
here upon earth do really descend : for a pendulum oscillating seconds in 
the latitude of Paris will be 3 Paris feet, and 8 lines 1 in length, as Mr. 
Hu.y veus has observed. And the space which a heavy body describes 
by falling in one second of time is to half the length of this pendulum in 
the duplicate ratio of the circumference of a circie to its diameter (as Mr. 
Htiy^ens has also shewn), and is therefore 15 Paris feet, I inch, 1 line J. 
And therefore the force by which the moon is retained in its orbit becomes, 
at the very surface of the earth, equal to the force of gravity which we ob 
serve in heavy bodies there. And therefore (by Rule I and II) the force by 
which the moon is retained in its orbit is that very same force which we 
commonly call gravity ; for, were gravity another force different from that, 
then bodies descending to the earth with the joint impulse of both forces 
would fall with a double velocity, and in the space of one second of time 
would describe 30^ Paris feet ; altogether against experience. 

This calculus is founded on the hypothesis of the earth s standing still ; 
for if both earth and moon move about the sun. and at the same time about 
their common centre of gravity, the distance of the centres of the moon and 
earth from one another will be 6(H semi-diameters of the earth ; as may 
be found by a computation from Prop. LX, Book I. 

SCHOLIUM. 

The demonstration of this Proposition may be more diffusely explained 
after the following manner. Suppose several moons to revolve about the 
earth, as in the system of Jupiter or Saturn : the periodic times of these 
moons (by the argument of induction) would observe the same law which 
Kepler found to obtain among the planets ; and therefore their centripetal 
forces would be reciprocally as the squares of the distances from the centre 
of the earth, by Prop. I, of this Book. Now if the lowest of these were 
very small, and were so near the earth as almost to touo the tops of the 
highest mountains, the centripetal force thereof, retaining it in its orb, 
would be very nearly equal to the weights of any terrestrial bodies that 
should be found upon the tops of those mountains, as may be known by 
the foregoing computation. Therefore if the same little moon should be 
deserted by its centrifugal force that carries it through its orb, and so be 



BOOK 111.] OF NATURAL PHILOSOPHY. 393 

lisabled from going onward therein, it would descend to the earth ; and 
that with the same velocity as heavy bodies do actually fall with upo-n the 
tops of those very mountains ; because of the equality of the forces that 
oblige them both to descend. And if the force by which that lowest moon 
would descend were different from gravity, and if that moon were to gravi 
tate towards the earth, as we find terrestrial bodies do upon the tops of 
mountains, it would then descend with twice the velocity, as being impel 
led by both these forces conspiring together. Therefore since both these 
forces, that is, the gravity of heavy bodies, and the centripetal forces of the 
moons, respect the centre of the earth, and are similar and equal between 
themselves, they will (by Rule I and II) have one and the same cause. And 
therefore the force which retains the moon in its orbit is that very force 
which we commonly call gravity ; because otherwise this little moon at the 
top of a mountain must either be without gravity, or fall twice as swiftly 
as heavy bodies are wont to do. 

PROPOSITION V. THEOREM V. 

Vhat the cir cum jovial planets gravitate towards Jupiter ; the circnntsat- 
urnal towards Saturn ; the circumsolar towards the sun ; and by t/ie 
forces of their gravity are drawn off from rectilinear motions, and re 
tained in curvilinear orbits. 

For the revolutions of the circumjovial planets about Jupiter, of the 
circumsaturnal about Saturn, and of Mercury and Venus, and the other 
circumsolar planets, about the sun, are appearances of the same sort with 
the revolution of the moon about the earth ; and therefore, by Rule II, 
must be owing to the same sort of causes ; especially since it has been 
demonstrated, that the forces upon which those revolutions depend tend to 
the centres of Jupiter, of Saturn, and of the sun ; and that those forces, in 
receding from Jupiter, from Saturn, and from the sun, decrease in the same 
proportion, and according to the same law, as the force of gravity does in 
receding from the earth. 

COR. 1. There is, therefore, a power of gravity tending to all the plan 
ets ; for, doubtless, Venus, Mercury, and the rest, are bodies of the same 
sort with Jupiter and Saturn. And since all attraction (by Law III) is 
mutual, Jupiter will therefore gravitate towards all his own satellites, Sat 
urn towards his, the earth towards the moon, and the sun towards all the 
primary planets. 

COR. 2. The force of gravity which tends to any one planet is re 
ciprocally as the square of the distance of places from that planet s 
centre. 

COR. 3. All the planets do mutually gravitate towards one another, by 
Cor. 1 and 2. And hence it is that Jupiter and Saturn, when near their 



394 THE MATHEMATICAL PRINCIPLES [BOOK III 

conjunction; by their mutual attractions sensibly disturb each other s ?n> 
tions. So the sun disturbs the motions of the moon ; and both sun ini 
moon disturb our sea, as we shall hereafter explain. 

SCHOLIUM. 

The force which retains the celestial bodi in their orbits has been 
hitherto called centripetal force; but it being now made plain that it can 
be no other than a gravitating force, we shall hereafter call it gravity. 
For the cause of that centripetal force which retains the moon in its orbit 
will extend itself to all the planets, by Rule I, II, and IV. 

PROPOSITION VI. THEOREM VI. 

That all bodies gravitate towards every planet ; and that the weights of 
bodies towards any the same planet, at equal distances from the centre 
of the planet, are proportional to the quantities of matter which they 
severally contain. 

It has been, now of a long time, observed by others, that all sorts of 
heavy bodies (allowance being made for the inequality of retardation which 
they suffer from a small power of resistance in the air) descend to the 
earth from equal heights in equal times; and that equality of times we 
may distinguish to a great accuracy, by the help of pendulums. I tried the 
thing in gold, silver, lead, glass, sand, eommpn salt, wood, water, and wheat. 
I provided two wooden boxes, round and equal : I filled the one with wood, 
and suspended an equal weight of gold (as exactly as I could) in the centre 
of oscillation of the other. The boxes hanging by equal threads of 11 feet 
made a couple of pendulums perfectly equal in weight and figure, and 
equally receiving the resistance of the air. And, placing the one by the 
other, I observed them to play together forward and backward, for a long 
time, wi h equal vibrations. And therefore the quantity of matte* : n the 
gold (by Cor. 1 and 6, Prop. XXIV, Book II) was to the quantity ot mat 
ter in the wood as the action of the motive force (or vis tnotrix) upon all 
the gold to the action of the same upon all the wood ; that is, as the weight 
of the one to the weight of the other : and the like happened in the other 
bodies. By these experiments, in bodies of the same weight, 1 could man 
ifestly have discovered a difference of matter less than the thousandth part 
of the whol^, had any such been. But, without all doubt, the nature of 
gravity towards the planets is the same as towards the earth. For, should 
we imagine our terrestrial bodies removed to the orb of the moon, and 
there, together with the moon, deprived of all motion, to be let go, so as to 
fall together towards the earth, it is certain, from what we have demonstra 
ted before, that, in equal times, they would describe equal spaces with the 
moon, and of consequence are to the moon, in quantity of matter, as their 
weights to its weight. Moreover, since the satellites of Jupiter perform 



HOOK ill.] or NVTURAL PHILOSOPHY, 395 

their revolutions in times which observe the sesquiphiate pr portion ol 
their distances from Jupiter s centre, their accelerative gravities towards 
Jupiter will be reciprocally as the squares of their distances from Jupiter s 
centre; that is, equal, at equal distances. And, therefore, these satellites, 
if supposed to fall towards Jupiter from equal heights, would describe equal 
spaces in equal times, in like manner as heavy bodies do on our earth. 
And, by the same argument, if the circumsolar planets were supposed to be 
let fall at equal distances from the sun, they would, in their descent towards 
the sun, describe equal spaces in equal times. But forces which equally 
accelerate unequal bodies must be as those bodies : that is to sa_y, the weights 
;f the planets towards the sun must be as their quantities of matter, 
further, that the weights of Jupiter and of his satellites towards the sun 
are proportional to the several quantities of their matter, appears from the 
exceedingly regular motions of the satellites (by Cor. 3, Prop. LXV, Book 
1). For if some of those bodies were more strongly attracted to the sun in 
proportion to their quantity of matter than others, the motions of the sat 
ellites would be disturbed by that inequality of attraction (by Cor.^, Prop. 
LXV, Book I). If, at equal distances from the sun, any satellite, in pro 
portion to the quantity of its matter, did gravitate towards the sun with a 
force greater than Jupiter in proportion to his, according to any given pro 
portion, suppose of d to e ; then the distance between the centres of the sun 
and of the satellite s orbit would be always greater than the distance be 
tween the centres of the sun and of Jupiter nearly in the subduplicate of 
that proportion : as by some computations I have found. And if the sat 
ellite did gravitate towards the sun with a force, lesser in the proportion of e 
to d, the distance of the centre of the satellite s orb from the sun would be 
less than the distance of the centre of Jupiter from the sun in the subdu 
plicate of the same proportion. Therefore if, at equal distances from the 
sun, the accelerative gravity of any satellite towards the sun were greater 
or less than the accelerative gravity of Jupiter towards the sun but by one T oV 7 
part of the whole gravity, the distance of the centre of the satellite s orbit 
from the sun would be greater or less than the distance of Jupiter from the 
sun by one ^oVo P art of the whole distance; that is, by a nf h part of the 
distance of the utmost satellite from the centre of Jupiter ; an eccentricity 
of the orbit which would be very sensible. But the orbits of the satellites 
are concentric to Jupiter, and therefore the accelerative gravities of Jupiter, 
and of all its satellites towards the sun, are equal among themselves. And 
by the same argument, the weights of Saturn and of his satellites towards 
the sun, at equal distances from the sun, are as their several quantities of 
matter ; and the weights of the moon and of the earth towards the sun are 
either none, or accurately proportional to the masses of matter which they 
contain. But some they are, by Cor. 1 and 3, Prop. V. 

But further ; the weights of all the parts of every planet f awards any other 



396 THE MATHEMATICAL PRINCIPLES [BOOK II] 

planet are one to another as the matter in the several parts; for if some 
parts did gravitate more, others less, than for the quantity of their matter, 
then the whole planet, according to the sort of parts with which it most 
abounds, would gravitate more or less than in proportion to the quantity of 
matter in the whole. Nor is it of any moment whether these parts are 
external or internal ; for if, for example, we should imagine the terrestrial 
bodies with us to be raised up to the orb of the moon, to be there compared 
with its body : if the weights of such bodies were to the weights of the ex 
ternal parts of the moon as the quantities of matter in the one and in the 
other respectively but to the weights of the internal parts in a greater or 
less proportion, then likewise the weights of those bodies would be to the 
weight of the whole moon in a greater or less proportion; against what 
we have shewed above. 

COR. 1. Hence the weights of bodies do not depend upon their forms 
and textures ; for if the weights could be altered with the forms, they 
would be greater or less, according to the variety of forms, in equal matter ; 
altogether against experience. 

COR. 2. Universally, all bodies about the earth gravitate towards the 
earth ; and the weights of all, at equal distances from the earth s centre. 
are as the quantities of matter which they severally contain. This is the 
quality of all bodies within the reach of our experiments ; and therefore 
(by Rule III) to be affirmed of all bodies whatsoever. If the ather, or anj 
other body, were either altogether void of gravity, or were to gravitate lesr 
in proportion to its quantity of matter, then, because (according to Aris 
totle, Des Carles, and others) there is no difference betwixt that and other 
bodies but in mere form of matter, by a successive change from form to 
form, it might be changed at last into a body of the same condition with 
those which gravitate most in proportion to their quantity of matter ; and, 
on the other hand, the heaviest bodies, acquiring the first form of that 
body, might by degrees quite lose their gravity. And therefore the weights 
would depend upon the forms of bodies, and with those forms might be 
changed : contrary to what was proved in the preceding Corollary. 

COR. 3. All spaces are not equally full; for if all spaces were equally 
full, then the specific gravity of the fluid which fills the region of the air, 
on account of the extreme density of the matter, would fall nothing short 
of the specific gravity of quicksilver, or gold, or any other the most dense 
body ; and, therefore, neither gold, nor any other body, could descend in 
air ; for bodies do not descend in fluids, unless they are specifically heavier 
than the fluids. And if the quantity of matter in a given space can, by 
any rarefaction, be diminished, what should hinder a diminution to 
infinity ? 

COR. 4. If all the solid particles of all bodies are of the same density, 
nor can be rarefied without pores, a void, space, or -acuum must be granted 



BOOK Ill.J OF NATURAL PHILOSOPHY. 397 

By bodies of the same density, I mean those whose vires inertia are in the 
proportion of their bulks. 

COR. 5. The power of gravity is of a different nature from the power of 
magnetism ; for the magnetic attraction is not as the matter attracted. 
Some bodies are attracted more by the magnet ; others less ; most bodies 
not at all. The power of magnetism in one and the same body may be 
increased and diminished ; and is sometimes far stronger, for the quantity 
of matter, than the power of gravity ; and in receding from the magnet 
decreases not in the duplicate but almost in the triplicate proportion of the 
distance, as nearly as I could judge from some rude observations. 

PROPOSITION VII. THEOREM VII. 

That there is a power of gravity tending to all bodies, proportional to 

the several quantities of matter which they contain. 

That all the planets mutually gravitate one towards another, we have 
proved before ; as well as that the force of gravity towards every one of them, 
considered apart, is reciprocally as the square of the distance of places from 
the centre of the planet. And thence (by Prop. LXIX, Book I, and its 
Corollaries) it follows, that the gravity tending towards all the planets is 
proportional to the matter which they contain. 

Moreover, since all the parts of any planet A gravitate towards any 
other planet B ; and the gravity of every part is to the gravity of the 
whole as the matter of the part to the matter of the whole ; and (by Law 
III) to every action corresponds an equal re-action ; therefore the planet B 
will, on the other hand, gravitate towards all the parts of the planet A ; 
and its gravity towards any one part will be to the gravity towards the 
whole as the matter of the part to the matter of the whole. Q.E.D. 

COR, 1. Therefore the force of gravity towards any whole planet arises 
from, and is compounded of, the forces of gravity towards all its parts. 
Magnetic and electric attractions afford us examples of this ; for all at 
traction towards the whole arises from the attractions towards the several 
parts. The thing may be easily understood in gravity, if we consider a 
greater planet, as formed of a number of lesser planets, meeting together in 
one globe ; for hence it would appear that the force of the whole must 
arise from the forces of the component parts. If it is objected, that, ac 
cording to this law, all bodies with us must mutually gravitate one to 
wards another, whereas no such gravitation any where appears, I answer, 
that since the gravitation towards these bodies is to the gravitation towards 
the whole earth as these bodies are to the whole earth, the gravitation to 
wards them must be far less than to fall under the observation of our senses. 

COR. 2. The force of gravity towards the several equal particles of any 
body is reciprocally as the square of the distance of places from the parti 
cles ; as appears from Cor. 3, Prop. LXXIV, Book I. 



39S THE MATHEMATICAL PRINCIPLES [BOOK III 



PROPOSITION VIII. THEOREM VIII. 

Tn two spheres mutually gravitating each towards the other, if tlie matter 
in places on all sides round about and equi-distant from the centres is 
similar, the weight of either sphere towards the other will be recipro 
cally as the square of the distance between their centres. 
After I had found that the force of gravity towards a whole planet did 
arise from and was compounded of the forces of gravity towards all its 
parts, and towards every one part was in the reciprocal proportion of the 
squares of the distances from the part, I was yet in doubt whether that re 
ciprocal duplicate proportion did accurately hold, or but nearly so, in the 
total force compounded of so many partial ones; for it might be that the 
proportion which accurately enough took place in greater distances should 
be wide of the truth near the surface of the planet, where the distances of 
the particles are unequal, and their situation dissimilar. But by the help 
of Prop. LXXV and LXXVI, Book I, and their Corollaries, I was at last 
satisfied of the truth of the Proposition, as it now lies before us. 

COR. 1. Hence we may find and compare together the weights of bodies 
towards different planets ; for the weights of bodies revolving in circles 
about planets are (by Cor. 2, Prop. IV, Book I) as the diameters of the 
circles directly, and the squares of their periodic times reciprocally ; and 
their weights at the surfaces of the planets, or at any other distances from 
their centres, are (by this Prop.) greater or less in the reciprocal duplicate 
proportion of the distances. Thus from the periodic times of Venus, re 
volving about the sun, in 224 <J . 16f h , of the utmost circumjovial satellite 
revolving about Jupiter, in 16 . 10 -?/. ; of the Huygenian satellite about 
Saturn in 15 d . 22f h . ; and of the moon about the earth in 27 d . 7 h . 43 ; 
compared with the mean distance of Venus from the sun, and with the 
greatest heliocentric elongations of the outmost circumjovial satellite 
from Jupiter s centre, 8 16"; of the Huygenian satellite from the centre 
of Saturn, 3 4" ; arid of the moon from the earth, 10 33" : by computa 
tion I found that the weight of equal bodies, at equal distances from the 
centres of the sun, of Jupiter, of Saturn, and of the earth, towards the sun, 
Jupiter, Saturn, and the earth, were one to another, as 1, T ^VT> ^oVr? an ^ 
___i___ respectively. Then because as the distances are increased or di 
minished, the weights are diminished or increased in a duplicate ratio, the 
weights of equal bodies towards the sun, Jupiter, Saturn, and the earth, 
at the distances 10000, 997, 791, and 109 from their centres, that is, at their 
very superficies, will be as 10000, 943, 529, and 435 respectively. How 
much the weights of bodies are at the superficies of the moon, will be 
shewn hereafter. 

COR. 2. Hence likewise we discover the quantity of matter in the several 



.BOOK II1.J OF NATURAL PHILOSOPHY. 39 ( .* 

planets; for their quantities of matter are as the forces of gravity at equai 
distances from their centres; that is, in the sun, Jupiter, Saturn, and the 
earth, as 1, TO FTJ a-oVr? anc ^ TeVaja respectively. If the parallax of the 
sun be taken greater or less than 10" 30 ", the quantity of matter in 
the earth must be augmented or diminished in the triplicate of that pro 
portion. 

COR. 3. Hence also we find the densities of the planets ; for (by Prop. 
LXXII, Book I) the weights of equal and similar bodies towards similar 
spheres are, at the surfaces of those spheres, as the diameters of the spheres 5 
and therefore the densities of dissimilar spheres are as those weights applied 
to the diameters of the spheres. But the true diameters of the Sun, .Jupi 
ter, Saturn, and the earth, were one to another as 10000, 997, 791, arid 
109; and the weights towards the same as 10000, 943, 529, and 435 re 
spectively ; and therefore their densities are as 100. 94|, 67, and 400. The 
density of the earth, which comes out by this computation, does not depend 
upon the parallax of the sun, but is determined by the parallax of the 
moon, and therefore is here truly defined. The sun, therefore, is a little 
denser than Jupiter, and Jupiter than Saturn, and the earth four times 
denser than the sun ; for the sun, by its great heat, is kept in a sort of 
a rarefied state. The moon is denser than the earth, as shall appear after 
ward. 

COR. 4. The smaller the planets are, they are, cccteris parilms, of so 
much the greater density ; for so the powers of gravity on their several 
surfaces come nearer to equality. They are likewise, cccteris paribiis, of 
the greater density, as they are nearer to the sun. So Jupiter is more 
dense than Saturn, and the earth than Jupiter ; for the planets were to be 
placed at different distances from the sun, that, according to their degrees 
of density, they might enjoy a greater or less proportion to the sun s heat. 
Our water, if it were removed as far as the orb of Saturn, would be con 
verted into ice, and in the orb of Mercury would quickly fly away in va 
pour ; for the light of the sun, to which its heat is proportional, is seven 
times denser in the orb of Mercury than with us : and by the thermometer 
I have found that a sevenfold heat of our summer sun will make water 
boil. Nor are we to doubt that the matter of Mercury is adapted to its 
heat, and is therefore more dense than the matter of our earth ; since, in a 
denser matter, the operations of Nature require a stronger heat. 

PROPOSITION IX. THEOREM IX. 

That the force of gravity, considered downward from t/ie surface 
of the planets decreases nearly in the proportion of the distances from 
their centres. 

If the matter of the planet were of an uniform density, this Proposi 
tion would be accurately true (by Prop. LXXIII. Book I). The error, 



100 THE MATHEMATICAL PRINCIPLES [BOOK III 

therefore, can be no greater than what may arise from the inequality of 
the density. 

PROPOSITION X. THEOREM X. 

That the motions of the planets in the heavens may subsist an exceedingly 

long time. 

In the Scholium of Prop. XL, Book II, I have shewed that a globe of 
water frozen into ice, and moving freely in our air, in the time that it would 
describe the length of its semi-diameter, would lose by the resistance of the 
air 3 \6 part of its motion; and the same proportion holds nearly in all 
globes, how great soever, and moved with whatever velocity. But that our 
globe of earth is of greater density than it would be if the whole 
consisted of water only, I thus make out. If the whole consisted of 
water only, whatever was of less density than water, because of its Ivss 
specific gravity, would emerge and float above. And upon this account, if 
a globe of terrestrial matter, covered on all sides with water, was less dense 
than water, it would emerge somewhere ; and, the subsiding water falling 
back, would be gathered to the opposite side. And such is the condition 
of our earth, which in a great measure is covered with seas. The earth, if 
it was not for its greater density, would emerge from the seas, and, accord 
ing to its degree of levity, would be raised more or less above their surface, 
the water of the seas flowing backward to the opposite side. By the same 
argument, the spots of the sun, which float upon the lucid matter thereof. 
are lighter than that matter ; and, however the planets have been formed 
while they were yet in fluid masses, all the heavier matter subsided to the 
centre. Since, therefore, the common matter of our earth on the surface 
thereof is about twice as heavy as water, and a little lower, in mines, is 
found about three, or four, or even five times more heavy, it is probable that 
the quantity of the whole matter of the earth may be five or six times 
greater than if it consisted all of water ; especially since I have before 
shewed that the earth is about four times more dense than Jupiter. If, 
therefore, Jupiter is a little more dense than water, in the space of thirty 
days, in which that planet describes the length of 459 of its semi-diame 
ters, it would, in a medium of the same density Avith our air, lose almost a 
tenth part of its motion. But since the resistance of mediums decreases 
in proportion to their weight or density, so that water, which is 13| times 
lighter than quicksilver, resists less in that proportion ; and air, which is 
860 times lighter than water, resists less in the same proportion ; therefore 
in the heavens, where the weight of the medium in which the planets move 
is immensely diminished, the resistance will almost vanish. 

It is shewn in the Scholium of Prop. XXII, Book II, that at the height 
of 200 miles above the earth the air is more rare than it is at the super 
ficies of the earth in the ratio of 30 to 0,0000000000003999, or as 



BOOK III.] OF NATURAL PHILOSOPHY. 401 

75000000000000 to 1 nearly. And hence the planet Jupiter, revolving in 
a medium of the same density with that superior air, would not lose by the 
resistance of the medium the 1000000th part of its motion in 1000000 
years. In the spaces near the earth the resistance is produced only by the 
air, exhalations, and vapours. When these are carefully exhausted by the 
air-pump from under the receiver, heavy bodies fall within the receiver with 
perfect freedom, and without the le.ist sensible resistance: gold itself, and 
the lightest down, let fall together, will descend with equal velocity; and 
though they fall through a space of four, six, and eight feet, they will come 
to the bottom at the same time; as appears from experiments. And there 
fore the celestial regions being perfectly void of air and exhalations, the 
planets and comets meeting no sensible resistance in those spaces will con 
tinue their motions through them for an immense tract of time. 

HYPOTHESIS I. 

That the centre of the system of the world is immovable. 
This is acknowledged by all, while some contend that the earth, 
others that the sun, is fixed in that centre. Let us see what may from 
hence follow. 

PROPOSITION XL THEOREM XI. 

That the common, centre of gravity of the earth, the sun, and all the 

planets, is immovable. 

For (by Cor. 4 of the Laws) that centre either is at rest, or moves uni 
formly forward in a right line ; but if that centre moved, the centre of the 
world would move also, against the Hypothesis. 

PROPOSITION XII. THEOREM XII. 

That the sun is agitated by a perpetual motion, but never recedes jar 

from the common, centre of gravity of all the planets. 
For since (by Cor. 2, Prop. VIII) the quantity of matter in the sun is to 
the quantity of matter in Jupiter as 1067 to 1 ; and the distance of Jupi 
ter from the sun is to the semi-diameter of the sun in a proportion but a 
small matter greater, the common centre of gravity of Jupiter and the sun 
will fall upon a point a little without the surface of the sun. By the same 
argument, since the quantity of matter in the sun is to the quantity of 
matter in Saturn as 3021 to 1, and the distance of Saturn from the sun is 
to the semi-diameter of the sun in a proportion but a small matter less, 
the common centre of gravity of Saturn and the sun will fall upon a point 
a little within the surface of the sun. And, pursuing the principles of this 
computation, we should find that though the earth and all the planets were 
placed on one side of the sun, the distance of the common centre of gravity 
of all from the centre of the sun would scarcely amount to one diameter of 

26 



102 THE MATHEMATICAL PRINCIPLES [BOOK III 

the sun. In other cases, the distances of those centres are always less : and 
therefore, since that centre of gravity is in perpetual rest, the sun, accord 
ing to the various positions of the planets, must perpetually be moved every 
way, but will never recede far from that centre. 

Con. Hence the common centre of gravity of the earth, the sun, and all 
the planets, is to be esteemed the centre of the world ; for since the earth, 
the sun, and all the planets, mutually gravitate one towards another, and 
are therefore, according to their powers of gravity, in perpetual agitation, 
as the Laws of Motion require, it is plain that their moveable centres can 
not be taken for the immovable centre of the world. If that body were to 
be placed in the centre, towards which other bodies gravitate most (accord 
ing to common opinion), that privilege ought to be allowed to the sun; but 
since the sun itself is moved, a fixed point is to be chosen from which the 
centre of the sun recedes least, and from which it would recede yet 
less if the body of the sun were denser and greater, and therefore less apt 
to be moved. 

PROPOSITION XIII. THEOREM XIII. 

The planets move in ellipses tvhicli have their common focus in the centre 
of the sini ; and, by radii drawn, to tJtat centre, they describe areas pro 
portion al to the times of description. 

We have discoursed above of these motions from the Phenomena. Now 
that we know the principles on which they depend, from those principles 
we deduce the motions of the heavens a priori. Because the weights of 
the planets towards the sun are reciprocally as the squares of their distan 
ces from the sun s centre, if the sun was at rest, and the other planets did 
not mutually act one upon another, their orbits would be ellipses, having 
the sun in their common focus; and they would describe areas proportional 
to the times of description, by Prop. I and XI, and Cor. 1, Prop. XIII, 
Book I. But the mutual actions of the planets one upon another are so 
very small, that they may be neglected ; and by Prop. LXVI, Book I, they 
less disturb the motions of the planets around the sun in motion than if 
those motions were performed about the sun at rest. 

It is true, that the action of Jupiter upon Saturn is not to be neglected; 
for the force of gravity towards Jupiter is to the force of gravity towards 
the sun (at equal distances, Cor. 2, Prop. VIII) as 1 to 1067; and therefore 
in the conjunction of Jupiter and Saturn, because the distance of Saturn 
from Jupiter is to the distance of Saturn from the sun almost as 4 to 9, the 
gravity of Saturn towards Jupiter will be to the gravity of Saturn towards 
the sun as 81 to 16 X 1067; or, as 1 to about 21 1. And hence arises a 
perturbation of the orb of Saturn in every conjunction of this planet with 
Tupiter, so sensible, that astronomers are puzzled with it. As the planet 



BOOK III.] OF NATURAL PHILOSOPHY. 403 

is differently situated in these conjunctions, its eccentricity is sometimes 
augmented, sometimes diminished; its aphelion is sometimes carried for 
ward, sometimes backward, and its mean motion is by turns accelerated and 
retarded ; yet the whole error in its motion about the sun, though arising 
from so great a force, may be almost avoided (except in the mean motion) 
by placing the lower focus of its orbit in the common centre of gravity of 
Jupiter and the sun (according to Prop. LXVII, Book I), and therefore that 
error, when it is greatest, scarcely exceeds two minutes ; and the greatest 
error in the mean motion scarcely exceeds two minutes yearly. But in the 
conjunction of Jupiter and Saturn, the accelerative forces of gravity of the 
sun towards Saturn, of Jupiter towards Saturn, and of Jupiter towards the 

sun, are almost as 16, 81, and - ~o^~~ > or 156609: and therefore 

the difference of the forces of gravity of the sun towards Saturn, and of 
Jupiter towards Saturn, is to the force of gravity of Jupiter towards the 
sun as 65 to 156609, or as 1 to 2409. But the greatest power of Saturn 
to disturb the motion of Jupiter is proportional to this difference; and 
therefore the perturbation of the orbit of Jupiter is much less than that of 
Saturn s. The perturbations of the other orbits are yet far less, except that 
the orbit of the earth is sensibly disturbed by the moon. The common 
centre of gravity of the earth and moon moves in an ellipsis about the sun 
in the focus thereof, and, by a radius drawn to the sun, describes areas pro 
portional to the times of description. But the earth in the mean time by 
a menstrual motion is revolved about this common centre. 

PROPOSITION XIV. THEOREM XIV. 

The aphelions and nodes of the orbits of the planets are fixed. 

The aphelions are immovable by Prop. XI, Book I ; and so are the 
planes of the orbits, by Prop. I of the same Book. And if the planes are 
fixed, the nodes must be so too. It is true, that some inequalities may 
arise from the mutual actions of the planets and comets in their revolu 
tions ; but these will be so small, that they may be here passed by. 

COR. 1. The fixed stars are immovable, seeing they keep the same posi 
tion to the aphelions and nodes of the planets. 

COR. 2. And since these stars are liable to no sensible parallax from the 
annual motion of the earth, they can have no force, because of their im 
mense distance, to produce any sensible effect in our system. Not to 
mention that the fixed stars, every where promiscuously dispersed in the 
heavens, by their contrary attractions destroy their mutual actions, by 
Prop. LXX, Book I. 

SCHOLIUM. 

Since the planets near the sun (viz. Mercury, Venus, the Earth, and 



404 THE MATHEMATICAL PRINCIPLES [B .-OK IIL 

Mars) are so small that they can act with but little force upon each other, 
therefore their aphelions and nodes must be fixed, excepting in so far as 
they are disturbed by the actions of Jupiter and Saturn, and other higher 
bodies. And hence we may find, by the theory of gravity, that their aphe 
lions move a little in consequentw, in respect of the fixed stars, and that 
in the sesqui plicate proportion of their several distances from the sun. So 
that if the aphelion of Mars, in the space of a hundred years, is carried 
33 20" in consequent-la, in respect of the fixed stars, the aphelions of the 
Earth, of Venus, and of Mercury, will in a hundred years be carried for 
wards 17 40", 10 53 , and 4 16", respectively. But these motions are 
so inconsiderable, that we have neglected them in this Proposition, 

PROPOSITION XV. PROBLEM I. 

To find the principal diameters <>f the orbits of the planets. 
They are to be taken in the sub-sesquiplicate proportion of the periodic 
times, by Prop. XV, Book I, and then to be severally augmented in the 
proportion of the sum of the masses of matter in the sun and each planet 
to the first of two mean proportionals betwixt that sum and the quantity of 
matter in the sun, by Prop. LX, Book I. 

PROPOSITION XVI. PROBLEM II. 

To find the eccentricities and aphelions of the planets. 
This Problem is resolved by Prop. XVIII, Book I. 

PROPOSITION XVII. THEOREM XV. 

That the diurnal motions of the planets are uniform, and that the 

libration of the moon arises from its diurnal motion. 
The Proposition is proved from the first Law of Motion, and Cor. 22, 
Prop. LXVI, Book I. Jupiter, with respect to the fixed stars, revolves in 
9 1 . 5(5 ; Mars in 24 h . 39 ; Venus in about 23 h . ; the Earth in 23 1 . 56 ; the 
Sun in 25 1 days, and the moon in 27 days, 7 hours, 43 . These things 
appear by the Phasnomena. The spots in the sun s body return to the 
same situation on the sun s disk, with respect to the earth, in 27 days ; and 
therefore with respect to the fixed stars the sun revolves in about 25|days. 
But because the lunar day, arising from its uniform revolution about its 
axis, is menstrual, that is, equal to the time of its periodic revolution in 
its orb, therefore the same face of the moon w r ill be always nearly turned to 
the upper focus of its orb ; but, as the situation of that focus requires, will 
deviate a little to one side and to the other from the earth in the lower 
focus j and this is the libration in longitude ; for the libration in latitude 
arises from the moon s latitude, and the inclination of its axis to the plane 
of the ecliptic. This theory of the libration of the moon, Mr. N. Mercato* 



BOOK III.] OF NATURAL PHILOSOPHY. 4() 

in his Astronomy, published at the beginning of the year 1676. explained 
more fully out of the letters I sent him. The utmost satellite of Saturn 
eeems to revolve about its axis with a motion like this of the moon, respect 
ing Saturn continually with the same face; for in its revolution round 
Saturn, as often as it comes to the eastern part of its orbit, it is scarcel) 
visible, and generally quite disappears ; which is like to be occasioned by 
some spots in that part of its body, which is then turned towards the earth, 
as M. Cassini has observed. So also the utmost satellite of Jupiter seema 
to revolve about its axis with a like motion, because in that part of its body 
which is turned from Jupiter it has a spot, which always appears as if it 
were in Jupiter s own body, whenever the satellite passes between Jupiter 
and our eye. 

PROPOSITION XVIII. THEOREM XVI. 

That the axes of the planets are less than the diameters drawn perpen 
dicular to the axes. 

The equal gravitation of the parts on all sides would give a spherical 
figure to the planets, if it was not for their diurnal revolution in a circle. 
By that circular motion it comes to pass that the parts receding from the 
axis endeavour to ascend about the equator ; and therefore if the matter is 
in a fluid state, by its ascent towards the equator it will enlarge the di 
ameters there, and by its descent to wards the poles it will shorten the axis. 
So the diameter of Jupiter (by the concurring observations of astronomers) 
is found shorter betwixt pole and pole than from east to west. And, by 
the same argument, if our earth was not higher about the equator than at 
the poles, the seas would subside about the poles, and, rising toward* Ikf 
equator, would lay all things there under water. 

PROPOSITION XIX. PROBLEM III 

To find the proportion of the axis of a planet to the dia meter j j*,rpen- 

dici/lar thereto. 

Our countryman, Mr. Norwood, measuring a distance of 005751 feet of 
London measure between London and YorA:, in 1635, and obs,-rvino- the 
difference of latitudes to be 2 28 , determined the measure of one degree 
to be 3671 96 feet of London measure, that is 57300 Paris toises. M 
Picart, measuring an arc of one degree, and 22 55" of the meridian be 
tween Amiens and Malvoisine, found an arc of one degree to be 57060 
Paris toises. M. Cassini, the father, measured the distance upon the me 
ridian from the town of Collionre in Roussillon to the Observatory of 
Pari; and his son added the distance from the Observatory to the Cita 
del of Dunkirk. The whole distance was 486156^ toises and the differ 
ence of the latitudes of Collionre and Dunkirk was 8 degrees, and 31 



106 THE MATHEMATICAL PRINCIPLES [BOOK 1IJ. 

llf". Hence an arc of one degree appears to be 57061 Paris toises. 
And from these measures we conclude that the circumference of the earth 
is 123249600, and its semi-diameter 19615800 Paris feet, upon the sup 
position that the earth is of a spherical figure. 

In the latitude of Paris a heavy body falling in a second of time de 
scribes 15 Paris feet, 1 inch, 1 J line, as above, that is, 2173 lines J. The 
weight of the body is diminished by the weight of the ambient air. Let 
us suppose the weight lost thereby to be TT ^o-o- P ar ^ ^ ^he w hole weight ; 
then that heavy body falling in, vacua will describe a height of 2174 lines 
in one second of time. 

A body in every sidereal day of 23 1 . 56 4" uniformly revolving in a 
circle at the distance of 19615SOO feet from the centre, in one second oi 
time describes an arc of 1433,46 feet ; the versed sine of which is 0,0523656 1 
feet, or 7,54064 lines. And therefore the force with which bodies descend 
in the latitude of Paris is to the centrifugal force of bodies in the equator 
arising from the diurnal motion of the earth as 2174 to 7,54064. 

The centrifugal force of bodies in the equator is to the centrifugal force 
with which bodies recede directly from the earth in the latitude of Parin 
48 50 10" in the duplicate proportion of the radius to the cosine of the 
latitude, that is, as 7,54064 to 3,267. Add this force to the force with 
which bodies descend by their weight in the latitude of Paris, and a body, 
in the latitude of Paris, falling by its whole undiminished force of gravity, 
in the time of one second, will describe 2177,267 lines, or 15 Paris feet, 
1 inch, and 5,267 lines. And the total force of gravity in that latitude 
will be to the centrifugal force of bodies in the equator of the earth as 
2177,267 to 7,54064, or as 289 to 1. 

Wherefore if APBQ, represent the figure of the 
earth, now no longer spherical, but generated by the 
rotation of an ellipsis about its lesser axis PQ, ; and 
ACQqca a canal full of water, reaching from the pole 
Qq to the centre Cc, and thence rising to the equator 
Art ; the weight of the water in the leg of the canal 
ACca will be to the weight of water in the other leg 
QCcq as 289 to 288, because the centrifugal force arising from the circu 
lar motion sustains and takes off one of the 289 parts of the weight (in the 
one leg), and the weight of 288 in the other sustains the rest. But by 
computation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matter 
of the earth was all uniform, and without any motion, and its axis PQ, 
were to the diameter AB as 100 to 101, the force of gravity in the 
place Q towards the earth would be to the force of gravity in the same 
place Q towards a sphere described about the centre C with the radius 
PC, or QC, as 126 to 125. And, by the same argument, the force of 
gravity in the place A towards the spheroid generated by the rotation of 




BOOK III.] OF NATURAL PHILOSOPHY. 407 

the ellipsis APBQ, about the axis AI3 is to the force of gravity in the 
same place A, towards the sphere described about the centre C with the 
radius AC, as 125 to 126. But the force of gravity in the place A to 
wards the earth is a mean proportional betwixt the forces of gravity to 
wards the spheroid and this sphere; because the sphere, by having its di 
ameter PQ, diminished in the proportion of 101 to 100, is transformed into 
the figure of the earth ; and this figure, by having a third diameter per 
pendicular to the two diameters AB and PQ, diminished in the same pro 
portion, is converted into the said spheroid ; and the force of gravity in A, 
in either case, is diminished nearly in the same proportion. Therefore the 
force of gravity in A towards the sphere described about the centre C with 
the radius AC, is to the force of gravity in A towards the earth as 126 to 
1251. And the force of gravity in the place Q towards the sphere de 
scribed about the centre C with the radius QC, is to the force of gravity 
in the place A towards the sphere described about the centre C, with the 
radius AC, in the proportion of the diameters (by Prop. LXXII, Book I), 
that is, as 100 to 101. If, therefore, we compound those three proportions 
126 to 125, 126 to 125|. and 100 to 101, into one, the force of gravity in 
the place Q towards the earth will be to the force of gravity in the place 
A towards the earth as 126 X 126 X 100 to 125 X 125| X 101 ; or as 
:>01 to 500. 

Now since (by Cor. 3, Prop. XCI, Book I) the force of gravity in either 
leg of the canal ACca, or QCcy, is as the distance of the places from the 
centre of the earth, if those legs are conceived to be divided by transverse., 
parallel, and equidistant surfaces, into parts proportional to the wholes, 
the weights of any number of parts in the one leg ACca will be to the 
weights of the same number of parts in the other leg as their magnitudes 
and the accelerative forces of their gravity conjunctly, that is, as 10 J to 
100, and 500 to 501. or as 505 to 501. And therefore if the centrifugal 
force of every part in the leg ACca, arising from the diurnal motion, was 
to the weight of the same part as 4 to 505, so that from the weight of 
every part, conceived to be divided into 505 parts, the centrifugal force 
might take off four of those parts, the weights would remain equal in each 
leg, and therefore the fluid would rest in an equilibrium. But the centri 
fugal force of every part is to the weight of the same part as 1 to 289 ; 
that is, the centrifugal force, which should be T y parts of the weight, is 
only |g part thereof. And, therefore, I say, by the rule of proportion, 
that if the centrifugal force j ^ make the height of the water in the leg 
ACca to exceed the height of the water in the leg QCcq by one T | part 
of its whole height, the centrifugal force -^jj will make the excess of the 
height in the leg ACca only ^{^ part of the height of the water in the 
other leg QCcq ; and therefore the diameter of the earth at the equator, is 
to its diameter from pole to pole as 230 to 229. And since the mean semi- 



108 



THE MATHEMATICAL PRINCIPLES 



[BooK III. 



diameter of the earth, according to PicarVs mensuration, is 19615800 
Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth 
will be higher at the equator than at the poles by 85472 feet, or 17^- 
miles. And its height at the equator will be about 19658600 feet, and at 
the poles 19573000 feet. 

If, the density and periodic time of the diurnal revolution remaining the 
same, the planet was greater or less than the earth, the proportion of the 
centrifugal force to that of gravity, and therefore also of the diameter be 
twixt the poles to the diameter at the equator, would likewise remain the 
game. But if the diurnal motion was accelerated or retarded in any pro 
portion, the centrifugal force would be augmented or diminished nearly in 
the same duplicate proportion ; and therefore the difference of the diame 
ters will be increased or diminished in the same duplicate ratio very nearly. 
And if the density of the planet was augmented or diminished in any pro 
portion, the force of gravity tending towards it would also be augmented 
or diminished in the same proportion : and the difference of the diameters 
contrariwise would be diminished in proportion as the force of gravity is 
augmented, and augmented in proportion as the force of gravity is dimin 
ished. Wherefore, since the earth, in respect of the fixed stars, revolves in 
23 h . 56 , but Jupiter in 9 h . 56 , and the squares of their periodic times are 
as 29 to 5, and their densities as 400 to 94 , the difference of the diameters 

29 400 1 

of Jupiter will be to its lesser diameter as X ^^ X ^Tm to 1; or as 1 to 

9 f, nearly. Therefore the diameter of Jupiter from east to west is to its 
diameter from pole to pole nearly as 10 to 9|-. Therefore since its 
greatest diameter is 37", its lesser diameter lying between the poles will 
be 33" 25" . Add thereto about 3 for the irregular refraction of light, 
and the apparent diameters of this planet will become 40 and 36" 25" ; 
which are to each other as 11 -j to 10^, very nearly. These things are so 
upon the supposition that the body of Jupiter is uniformly dense. But 
now if its body be denser towards the plane of the equator than towards 
the poles, its diameters may be to each other as 12 to 11, or 13 to 12, or 
perhaps as 14 to 13. 

And Cassini observed in the year 1691, that the diameter of Jupiter 
reaching from east to west is greater by about a fifteenth part than the 
other diameter. Mr. Pound with his 123 feet telescope, and an excellent 
micrometer, measured the diameters of Jupiter in the year 1719, and found 
them as follow. 



The Times. 


(jieatestdiam. 


Lesser diam. 


The diam. to each other. 


Day. Hours. 

January 28 6 
March 6 7 
March 9 7 
April 9 9 


I arts 
13,40 
13,12 
13.12 
X2.32 


I arts 

12,28 
12,20 
12,08 
11,48 


As 12 to 11 
13| to 12| 
12f to llf 
14 to 13d 



K HI. OF NATURAL PHILOSOPHY. 409 




So thut the theory agrees with the phenomena ; for the planets are more 
heated by the sun s rays towards their equators, and therefore are a lit fie 
more condensed by that heat than towards their poles. 

Moreover, that there is a diminution of gravity occasioned by the diur 
nal rotation of the earth, and therefore the earth rises higher there than it 
does at the poles (supposing that its matter is uniformly dense), will ap 
pear by the experiments of pendulums related under the following Propo 
sition. 

PROPOSITION XX. PROBLEM IV. 

To find and compare together the weights of bodies in the different re 

gions of our earth. 

Because the weights of the unequal legs of the canal 
of water ACQqca are equal ; and the weights of the 
parts proportional to the whole legs, and alike situated 
in them, are one to another as the weights of the P| 
wholes, and therefore equal betwixt themselves ; the 
weights of equal parts, and alike situated in the legs, 
will be reciprocally as the legs, that is, reciprocally as 
230 to 229. And the case is the same in all homogeneous equal bodies alike 
situated in the legs of the canal. Their weights are reciprocally as the legs, 
that is, reciprocally as the distances of the bodies from the centre of the earth. 
Therefore if the bodies are situated in the uppermost parts of the canals, or on 
the surface of the earth, their weights will be one to another reciprocally as 
their distances from the centre. And. by the same argument, the weights in 
all other places round the whole surface of the earth are reciprocally as the 
distances of the places from the centre ; and, therefore, in the hypothesis 
of the earth s being a spheroid are given in proportion. 

Whence arises this Theorem, that the increase of weight in passing from 
tne equator to the poles is nearly as the versed sine of double the latitude ; 
or, which comes to the same thinir, as the square of the right sine of the 
latitude ; and the arcs of the degrees of latitude in the meridian increase 
nearly in the same proportion. And, therefore, since the latitude of Paris 
is 48 50 , that of places under the equator 00 00 , and that of places 
under the poles 90 ; and the versed sines of double those arcs are 
11334,00000 and 20000, the radius being 10000 ; and the force of gravity 
at the pole is to the force of gravity at the equator as 230 to 229 ; and 
the excess of the force of gravity at the pole to the force of gravity at the 
equator as 1 to 229 ; the excess of the force of gravity in the latitude of 
Paris will be to the force of gravity at the equator as 1 X Htll to 229, 
or as 5667 to 2290000. And therefore the whole forces of gravity in 
those places will be one to the other as 2295667 to 2290000. Wherefore 
since the lengths of pendulums vibrating in equal times are as the forces of 



410 



THE MATHEMATICAL PRINCIPLES 



[BOOK III. 



gravity, and in the latitude of Paris, the length of a pendulum vibrating 
seconds is 3 Paris feet, and S lines, or rather because of the weight of 
the air, 8f lines, the length of a pendulum vibrating in the same time 
arider the equator will be shorter by 1,087 lines. And by a like calculus 
the following table is made. 



Latitude of 
the place. 


Length ol the 
pendulum 


iMeasure of one degree 
in the meridian. 


Deg. 


Feet Lines. 


Toises. 





3 


. 7,468 


56637 


5 


3 


. 7,482 


56642 


10 


3 


. 7,526 


56659 


15 


3 


. 7,596 


56687 


20 


3 


. 7,692 


56724 


25 


3 


. 7,812 


56769 


30 


3 


. 7,948 


56823 


35 


3 


. 8,099 


56882 


40 


3 


. 8,261 


56945 


1 


3 


. 8.294 


5695? 


2 


3 


. 8,327 


5697 1 


3 


3 


. 8,361 


56984 


4 


3 


. 8 ; 394 


56997 


45 


3 


. 8.428 


57010 


6 


3 


. 8,461 


57022 


7 


3 


. 8,494 


57035 


8 


3 


. 8,528 


57048 


9 


3 


. 8,561 


57061 


50 


3 


. 8,594 


57074 


55 


3 


. 8.756 


57137 


60 


3 


. 8^907 


57196 


65 


3 


. 9,044 


57250 


70 


3 


. 9,162 


57295 


75 


3 


. 9,258 


57332 


80 


3 


. 9,329 


57360 


85 


3 


. 9,372 


57377 


90 


3 


. 9,387 


57382 



By this table, therefore, it appears that the inequality of degrees is sc 
small, that the figure of the earth, in geographical matters, may be con 
sidered as spherical ; especially if the earth be a little denser towards the 
plane of the equator than towards the poles. 

Now several astronomers, sent into remote countries to make astronomical 
observations, have found that pendulum clocks do accordingly move slower 
near the equator than in our climates. And, first of all, in the year I 72, 
M. Richer took notice of it in the island of Cayenne ; for when, in the 
month of August, he was observing the transits of the fixed stars over the 
meridian, he found his clock to go slower than it ought in respect of the 
mean motion of the sun at the rate of 2 29" a day. Therefore, fitting up 
a simple pendulum to vibrate in seconds, which were measured by an ex 
cellent clock, he observed the length of that simple pendulum ; and this he 
did over and over every week for ten months together. And upon his re 
turn to France, comparing the length of that pendulum with the length 



iiJ.j OF NATURAL PHILOSOPHY. 411 

of the pendulum at Paris (which was 3 Paris feet and 8f lines), he found 
it shorter by 1 j line. 

Afterwards, our friend Dr. Halley, about the year 1677, arriving at the 
island of St. Helena, found his pendulum clock to go slower there than at 
Isondon without marking the difference. But he shortened the rod of 
his clock by more than the \ of an inch, or l line ; and to effect this, be 
cause the length of the screw at the lower end of the rod was riot sufficient, 
he interposed a wooden ring betwixt the nut and the ball. 

Then, in the year 1682, M. Varin and M. des Hayes found the length 
of a simple pendulum vibrating in seconds at the Royal Observatory of 
Paris to be 3 feet and S| lines. And by the same method in the island 
of Goree, they found the length of an isochronal pendulum to be 3 feet and 
6 1 lines, differing from the former by two lines. And in the same year, 
going to the islands of Guadeloupe and Martinico, they found that the 
length of an isochronal pendulum in those islands was 3 feet and 6^ lines. 

After this, M. Couplet, the son, in the month of July 1697, at the Royal 
Observatory of Paris, so fitted his pendulum clock to the mean motion of 
the sun, that for a considerable time together the clock agreed with the 
motion of the sun. In November following, upon his arrival at Lisbon, he 
found his clock to go slower than before at the rate of 2 13" in 24 hours. 
And next March coming to Paraiba, he found his clock to go slower than 
at Paris, and at the rate 4 12" in 24 hours ; and he affirms, that the pen 
dulum vibrating in seconds was shorter at Lisbon by 2 lines, and at Pa 
raiba, by 3 1 lines, than at Paris. He had done better to have reckoned 
those differences \\ and 2f : for these differences correspond to the differ 
ences of the times 2 13" and 4 12". But this gentleman s observations 
are so gross, that we cannot confide in them. 

In the following years, 1699, and 1700, M. des Hayes, making another 
voyage to America, determined that in the island of Cayenne and Granada 
the length of the pendulum vibrating in seconds was a small matter less 
than 3 feet and 6| lines ; that in the island of St. Christophers it was 
3 feet and 6f lines ; and in the island of St. Domingo 3 feet and 7 
lines. 

And in the year 1704, P. Feuille, at Puerto Bello in America, found 
that the length of the pendulum vibrating in seconds was 3 Paris feet, 
and only 5--^ lines, that is, almost 3 lines shorter than at Paris ; but the 
observation was faulty. For afterward, going to the island of Martinico. 
he found the length of the isochronal pendulum there 3 Paris feet and 
5 \ | lines. 

Now the latitude of Paraiba is 6 38 south ; that of Puerto Bello 9 
33 north ; and the latitudes of the islands Cayenne, Goree, Gaudaloupe } 
Martinico, Granada, St. Christophers, and St. Domingo, are respectively 
4 C 55 , 14 40", 15 00 , 14 44 , 12 06 , 17 19 , and 19 48 , north. An*J 



412 THE MATHEMATICAL PRINCIPLES [BOOK III 

the excesses of the length of the pendulum at Paris above the lengths of 
the isochronal pendulums observed in those latitudes are a little greater 
than by the table of the lengths of the pendulum before computed. And 
therefore the earth is a little higher under the equator than by the prece 
ding calculus, and a little denser at the centre than in mines near the sur 
face, unless, perhaps, the heats of the torrid zone have a little extended the 
length of the pendulums. 

For M. Picart has observed, that a rod of iron, which in frosty weather 
in the winter season was one foot long, when heated by lire, was lengthened 
into one foot and -]- line. Afterward M. de la Hire found that a rod of 
iron, which in the like winter season was 6 feet long, when exposed to the 
heat of the summer sun, was extended into 6 feet and f line. In the former 
case the heat was greater than in the latter ; but in the latter it was greater 
than the heat of the external parts of a human body ; for metals exposed 
to the summer sun acquire a very considerable degree of heat. But the rod 
of a pendulum clock is never exposed to the heat of the summer sun, nor 
ever acquires a heat equal to that of the external parts of a human body ; 
and, therefore, though the 3 feet rod of a pendulum clock will indeed be a 
little longer in the summer than in the winter season, yet the difference will 
scarcely amount to \ line. Therefore the total difference of the lengths of 
isochronal pendulums in different climates cannot be ascribed to the differ 
ence of heat ; nor indeed to the mistakes of the French astronomers. For 
although there is not a perfect agreement betwixt their observations, yet 
the errors are so small that they may be neglected ; and in this they all 
agree, that isochronal pendulums are shorter under the equator than 
at the Royal Observatory of Paris, by a difference not less than 1{ line, 
nor greater than 2| lines. By the observations of M. Richer, in the island 
of Cayenne, the difference was 1| line. That difference being corrected by 
those of M. des Hayes, becomes \\ line or l line. By the less accurate 
observations of others, the same was made about two lines. And this dis 
agreement might arise partly from the errors of the observations, partly 
from the dissimilitude of the internal parts of the earth, and the height of 
mountains ; partly from the different heats of the air. 

I take an iron rod of 3 feet long to be shorter by a sixth part of one line 
in winter time with us here in England than in the summer. Because of 
the great heats under the equator, subduct this quantity from the difference 
of one line and a quarter observed by M. Richer, and there will remain one 
line T V, which agrees very well with l T -oo ^ ne collected, by the theory a 
little before. M. Richer repeated his observations, made in the island of 
Cayenne, every week for ten months together, and compared the lengths of 
the pendulum which he had there noted in the iron rods with the lengths 
thereof which he observed in Prance. This diligence and care seems to 
have been wanting to the other observers. If this gentleman s observations 



BOOK I1I.J OF NATURAL PHILOSOPHY. 413 

are to be depended on, the earth is higher under the equator than at the 
poles, and that by an excess of about 17 miles; as appeared above by the 
theory. 

PROPOSITION XXI. THEOREM XVII. 

That the equinoctial points go backward, and that the axis of the earth, 
by a nutation in, every annual revolution, twice vibrates towards the 

ecliptic, and as often returns to its former position,. 
The proposition appears from Cor. 20, Prop. LXVI, Book I ; but 
that motion of nutation must be very small, and, indeed, scarcely per 
ceptible. 

PROPOSITION XXII. THEOREM XVIII. 

That all the motions of the ?noon, and all the inequalities of those motions, 

follow from the principles which we have laid down. 
That the greater planets, while they are carried about the sun, may in 
the mean time carry other lesser planets, revolving about them ; and that 
those lesser planets must move in ellipses which have their foci in the cen 
tres of the greater, appears from Prop. LXV, Book I. But then their mo 
tions will be several ways disturbed by the action of the sun, and they will 
suffer such inequalities as are observed in our moon. Thus our moon (by 
Cor. 2, 3, 4, and 5, Prop. LXVI, Book I) moves faster, and, by a radius 
drawn to the earth, describes an area greater for the time, and has its orbit 
less curved, and therefore approaches nearer to the earth in the syzygies 
than in the quadratures, excepting in so far as these effects are hindered by 
the motion of eccentricity ; for (by Cor. 9, Prop. LXVI, Book I) the eccen 
tricity is greatest when the apogeon of the moon is in the syzygies, and 
least when the same is in the quadratures ; and upon this account the pe- 
rigeon moon is swifter, and nearer to us, but the apogeon moon slower, 
arid farther from us, in the syzygies than in the quadratures. Moreover, 
the apogee goes forward, and the nodes backward ; and this is done not with 
a regular but an unequal motion. For (by Cor. 7 and 8, Prop. LXVI, 
Book I) the apogee goes more swiftly forward in its syzygies, more slowly 
backward in its quadratures; and, by the excess of its progress above its 
regress, advances yearly in consequentia. But, contrariwise, the nodes (by 
Cor. 11, Prop. LXVI, Book I) are quiescent in their syzygies, and go fastest 
back in their quadratures. Farther, the greatest latitude of the moon (by 
Cor. 10, Prop. LXVI, Book I) is greater in the quadratures of the moon 
than in its syzygies. And (by Cor. 6, Prop. LXVI, Book I) the mean mo 
tion of the moon is slower in the perihelion of the earth than in its aphelion. 
And these are the principal inequalities (of the moon) taken notice of by 
astronomers. 



414 THE MATHEMATICAL PRINCIPLES [BOOK III 

But there are yet other inequalities not observed by former astronomers, 
by which the motions of the moon are so disturbed, that to this day we 
have not been able to bring them under any certain rule. For the veloc 
ities or horary motions of the apogee and nodes of the moon, and their 
equations, as well as the difference betwixt the greatest eccentricity in the 
syzygics, and the least eccentricity in the quadratures, and that inequality 
which we call the variation, are (by Cor. 14, Prop. LXVI, Book I) in the 
course of the year augmented and diminished in the triplicate proportion 
of the sun s apparent diameter. And besides (by Cor. 1 and 2, Lem. 10, 
and Cor. 16, Prop. LXVI, Book I) the variation is augmented and 
diminished nearly in the duplicate proportion of the time between 
the quadratures. But in astronomical calculations, this inequality 
is commonly thrown into and confounded with the equation of the moon s 
centre. 

PROPOSITION XXI1L PROBLEM V. 

To derive the unequal motions of the satellites of Jupiter and Saturn 

from the motions of our moon. 

From the motions of our moon we deduce the corresponding motions of 
the moons or satellites of Jupiter in this manner, by Cor. 16, Prop. LXVI, 
Book I. The mean motion of the nodes of the outmost satellite of Jupiter 
is to the mean motion of the nodes of our moon in a proportion compound 
ed of the duplicate proportion of the periodic times of the earth about the 
sun to the periodic times of Jupiter about the sun, and the simple propor 
tion of the periodic time of the satellite about Jupiter to the periodic time 
of our moon about the earth ; and, therefore, those nodes, in the space of 
a hundred years, are carried 8 24 backward, or in antecedentia. The 
mean motions of the nodes of the inner satellites are to the mean motion of 
the nodes of the outmost as their periodic times to the periodic time of the 
former, by the same Corollary, and are thence given. And the motion of 
the apsis of every satellite in consequential is to the motion of its nodes in 
antecedentia as the motion of the apogee of our moon to the motion of its 
nodes (by the same Corollary), and is thence given. But the motions of 
the apsides thus found must be diminished in the proportion of 5 to 9, or 
of about 1 to 2, on account of a cause which I cannot here descend to ex 
plain. The greatest equations of the nodes, and of the apsis of every satel 
lite, are to the greatest equations of the nodes, and apogee of our moon re 
spectively, as the motions of the nodes and apsides of the satellites, in the 
time of one revolution of the former equations, to the motions of the nodes 
and apogee of our moon, in the time of one revolution of the latter equa 
tions. The variation of a satellite seen from Jupiter is to the variation of 
our moon in tne same proportion as the whole motions of their node? 



BOOK IIIJ OF NATURAL PHILOSOPHY. 415 

respectively during the times in which the satellite and our moon (after 
parting from) are revolved (again) to the sun, by the same Corollary ; and 
therefore in the outmost satellite the variation does not exceed 5" 12 ". 

PROPOSITION XXIV. THEOREM XIX. 

That the flax and reflux of the sea arise from the actions oj the sun 

and moon. 

By Cor. 19 and 20, Prop. LXVI, Book I, it appears that the waters of 
the sea ought twice to rise and twice to fall every day. as well lunar as solar ; 
and that the greatest height of the waters in the open and deep seas ought 
to follow the appulse of the luminaries t